An Efficient Numerical Approach for Nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin; Vedula, Prakash
2009-03-01
Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.
A quadrature based method of moments for nonlinear Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Otten, Dustin L.; Vedula, Prakash
2011-09-01
Fokker-Planck equations which are nonlinear with respect to their probability densities and occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, fermions and bosons can be challenging to solve numerically. To address some underlying challenges, we propose the application of the direct quadrature based method of moments (DQMOM) for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations (NLFPEs). In DQMOM, probability density (or other distribution) functions are represented using a finite collection of Dirac delta functions, characterized by quadrature weights and locations (or abscissas) that are determined based on constraints due to evolution of generalized moments. Three particular examples of nonlinear Fokker-Planck equations considered in this paper include descriptions of: (i) the Shimizu-Yamada model, (ii) the Desai-Zwanzig model (both of which have been developed as models of muscular contraction) and (iii) fermions and bosons. Results based on DQMOM, for the transient and stationary solutions of the nonlinear Fokker-Planck equations, have been found to be in good agreement with other available analytical and numerical approaches. It is also shown that approximate reconstruction of the underlying probability density function from moments obtained from DQMOM can be satisfactorily achieved using a maximum entropy method.
Study of Bunch Instabilities By the Nonlinear Vlasov-Fokker-Planck Equation
Warnock, Robert L.; /SLAC
2006-07-11
Instabilities of the bunch form in storage rings may be induced through the wake field arising from corrugations in the vacuum chamber, or from the wake and precursor fields due to coherent synchrotron radiation (CSR). For over forty years the linearized Vlasov equation has been applied to calculate the threshold in current for an instability, and the initial growth rate. Increasing interest in nonlinear aspects of the motion has led to numerical solutions of the nonlinear Vlasov equation, augmented with Fokker-Planck terms to describe incoherent synchrotron radiation in the case of electron storage rings. This opens the door to much deeper studies of coherent instabilities, revealing a rich variety of nonlinear phenomena. Recent work on this topic by the author and collaborators is reviewed.
Nonlinear Bayesian estimation via solution of the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Yoon, Jangho
2009-12-01
A general approach to optimal nonlinear filtering can be described by a recursive Bayesian approach. The key step in this approach is to determine the probability density function of the state vector conditioned on available measurements. However, an optimal solution to the Bayesian filtering problem can only be obtained exactly for a small class of problems such as linear and Gaussian cases. Therefore, in practice, approximate solutions, such as the extended Kalman filter, have been used. An optimal nonlinear filtering in a recursive Bayesian approach is a two-step process which consists of the prediction and the update process. In the update process, the priori conditional state probability density function (PDF) from the prediction process is updated through Bayes' rule using measurements from sensors. The prediction of conditional state PDF can be made by solving the Fokker-Planck equation (FPE) that governs the time-evolution the conditional state PDF. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with the exception of a few special cases. So far this estimation method has not been employed much in practice because of the high computational cost needed in solving the FPE numerically. In this dissertation, methods to improve the efficiency of the numerical method in solving the FPE are investigated to enhance the efficiency of the nonlinear filtering. Two finite difference methods, namely (i) the explicit forward method; and (ii) the alternating direction implicit (ADI) method, are used to solve the FPE numerically. Although the explicit forward method is much simpler to implement, the ADI method is preferred for its low computational cost. To reduce the computational cost further, as the first contribution of the dissertation, a moving domain scheme is developed to reduce the domain of integration required for solving the Fokker-Planck equation numerically. Simulation results show that the accuracy of the
Global existence for a nonlocal and nonlinear Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, Michael
2015-04-01
We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier, one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state.
Fokker-Planck equation analysis of randomly excited nonlinear energy harvester
NASA Astrophysics Data System (ADS)
Kumar, P.; Narayanan, S.; Adhikari, S.; Friswell, M. I.
2014-03-01
The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker-Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.
Trigger, S. A.; Ebeling, W.; Heijst, G. J. F. van; Litinski, D.
2015-04-15
The problems of high linear conductivity in an electric field, as well as nonlinear conductivity, are considered for plasma-like systems. First, we recall several observations of nonlinear fast charge transport in dusty plasma, molecular chains, lattices, conducting polymers, and semiconductor layers. Exploring the role of noise we introduce the generalized Fokker-Planck equation. Second, one-dimensional models are considered on the basis of the Fokker-Planck equation with active and passive velocity-dependent friction including an external electrical field. On this basis, it is possible to find the linear and nonlinear conductivities for electrons and other charged particles in a homogeneous external field. It is shown that the velocity dependence of the friction coefficient can lead to an essential increase of the electron average velocity and the corresponding conductivity in comparison with the usual model of constant friction, which is described by the Drude-type conductivity. Applications including novel forms of controlled charge transfer and non-Ohmic conductance are discussed.
NASA Astrophysics Data System (ADS)
Nevolin, V. I.
2003-04-01
We present a method for analyzing the characteristics of nonlinear detectors using the algorithms of first-order nonlinear differential equations. This method is based on numerical solutions of the Fokker-Planck-Kolmogorov (FPK) equations in the form of series of functions over Hermite-Chebyshev polynomials for both nonlinear systems and their linear counterparts. The results of the solutions for the linear case are extended to nonlinear systems in a recurrent way.
Fokker-Planck equation in mirror research
Post, R.F.
1983-08-11
Open confinement systems based on the magnetic mirror principle depend on the maintenance of particle distributions that may deviate substantially from Maxwellian distributions. Mirror research has therefore from the beginning relied on theoretical predictions of non-equilibrium rate processes obtained from solutions to the Fokker-Planck equation. The F-P equation plays three roles: Design of experiments, creation of classical standards against which to compare experiment, and predictions concerning mirror based fusion power systems. Analytical and computational approaches to solving the F-P equation for mirror systems will be reviewed, together with results and examples that apply to specific mirror systems, such as the tandem mirror.
Fractional Fokker-Planck equation for fractal media.
Tarasov, Vasily E
2005-06-01
We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space. PMID:16035878
Spectral Decomposition of a Fokker-Planck Equation at Criticality
NASA Astrophysics Data System (ADS)
Bologna, M.; Beig, M. T.; Svenkeson, A.; Grigolini, P.; West, B. J.
2015-07-01
The mean field for a complex network consisting of a large but finite number of random two-state elements, , has been shown to satisfy a nonlinear Langevin equation. The noise intensity is inversely proportional to . In the limiting case , the solution to the Langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point. When , the inverse power law is truncated by an exponential decay with rate , the evaluation of which is the main purpose of this article. An analytic/numeric approach is used to obtain the lowest-order eigenvalues in the spectral decomposition of the solution to the corresponding Fokker-Planck equation and its equivalent Schrödinger equation representation.
Problems with the linear q-Fokker Planck equation
NASA Astrophysics Data System (ADS)
Yano, Ryosuke
2015-05-01
In this letter, we discuss the linear q-Fokker Planck equation, whose solution follows Tsallis distribution, from the viewpoint of kinetic theory. Using normal definitions of moments, we can expand the distribution function with infinite moments for 0 ⩽ q < 1, whereas we cannot expand the distribution function with infinite moments for 1 < q owing to emergences of characteristic points in moments. From Grad's 13 moment equations for the linear q-Fokker Planck equation, the dissipation rate of the heat flux via the linear q-Fokker Planck equation diverges at 0 ⩽ q < 2/3. In other words, the thermal conductivity, which defines the heat flux with the spatial gradient of the temperature and the thermal conductivity, which defines the heat flux with the spacial gradient of the density, jumps to zero at q = 2/3, discontinuously.
The Fokker-Planck Equation with Absorbing Boundary Conditions
NASA Astrophysics Data System (ADS)
Hwang, Hyung Ju; Jang, Juhi; Velázquez, Juan J. L.
2014-10-01
We study the initial-boundary value problem for the Fokker-Planck equation in an interval with absorbing boundary conditions. We develop a theory of well-posedness of classical solutions for the problem. We also prove that the resulting solutions decay exponentially for long times. To prove these results we obtain several crucial estimates, which include hypoellipticity away from the singular set for the Fokker-Planck equation with absorbing boundary conditions, as well as the Hölder continuity of the solutions up to the singular set.
NASA Astrophysics Data System (ADS)
Ichiki, A.; Shiino, M.
2009-08-01
Phase transitions and effects of external noise on many-body systems are one of the main topics in physics. In mean-field coupled nonlinear dynamical stochastic systems driven by Brownian noise, various types of phase transitions including nonequilibrium ones may appear. A Brownian motion is a special case of Lévy motion and the stochastic process based on the latter is an alternative choice for studying cooperative phenomena in various fields. Recently, fractional Fokker-Planck equations associated with Lévy noise have attracted much attention and behaviors of systems with double-well potential subjected to Lévy noise have been studied intensively. However, most of such studies have resorted to numerical computation. We construct an analytically solvable model to study the occurrence of phase transitions driven by Lévy stable noise.
New exact solutions to the Fokker Planck Kolmogorov equation
NASA Astrophysics Data System (ADS)
Ünal, Gazanfer; Sun, Jian-Qiao
2008-12-01
A generalized Liouville theorem has been proven for Itô systems. This allows us to show that the conserved quantities of the deterministic part of the Itô systems lead to the solution of the Fokker-Planck-Kolmogorov equation. The results have been applied to a stochastic 3-species Lotka Volterra system and the semi-classical Jaynes-Cummings system.
NASA Astrophysics Data System (ADS)
Che, Rui; Huang, Wen; Li, Yao; Tetali, Prasad
2016-08-01
In 2012, Chow, Huang, Li and Zhou [7] proposed the Fokker-Planck equations for the free energy on a finite graph, in which they showed that the corresponding Fokker-Planck equation is a nonlinear ODE defined on a Riemannian manifold of probability distributions. Different choices for inner products result in different Fokker-Planck equations. The unique global equilibrium of each equation is a Gibbs distribution. In this paper we proved that the exponential rate of convergence towards the global equilibrium of these Fokker-Planck equations. The rate is measured by both the decay of the L2 norm and that of the (relative) entropy. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [7]. The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [7].
Quantum Fokker-Planck-Kramers equation and entropy production
NASA Astrophysics Data System (ADS)
de Oliveira, Mário J.
2016-07-01
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance.
State-space-split method for some generalized Fokker-Planck-Kolmogorov equations in high dimensions
NASA Astrophysics Data System (ADS)
Er, Guo-Kang; Iu, Vai Pan
2012-06-01
The state-space-split method for solving the Fokker-Planck-Kolmogorov equations in high dimensions is extended to solving the generalized Fokker-Planck-Kolmogorov equations in high dimensions for stochastic dynamical systems with a polynomial type of nonlinearity and excited by Poissonian white noise. The probabilistic solution of the motion of the stretched Euler-Bernoulli beam with cubic nonlinearity and excited by uniformly distributed Poissonian white noise is analyzed with the presented solution procedure. The numerical analysis shows that the results obtained with the state-space-split method together with the exponential polynomial closure method are close to those obtained with the Monte Carlo simulation when the relative value of the basic system relaxation time and the mean arrival time of the Poissonian impulse is in some limited range.
Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix
Schulze-Halberg, Axel
2012-10-15
We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schroedinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.
Fokker-Planck equation of Schramm-Loewner evolution.
Najafi, M N
2015-08-01
In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE(κ,ρc). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long- and short-time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE(κ,ρc) traces and show that it is related to the higher-order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented. PMID:26382350
Hypersonic expansion of the Fokker--Planck equation
Fernandez-Feria, R.
1989-02-01
A systematic study of the hypersonic limit of a heavy species diluted in a much lighter gas is made via the Fokker--Planck equation governing its velocity distribution function. In particular, two different hypersonic expansions of the Fokker--Planck equation are considered, differing from each other in the momentum equation of the heavy gas used as the basis of the expansion: in the first of them, the pressure tensor is neglected in that equation while, in the second expansion, the pressure tensor term is retained. The expansions are valid when the light gas Mach number is O(1) or larger and the difference between the mean velocities of light and heavy components is small compared to the light gas thermal speed. They can be applied away from regions where the spatial gradient of the distribution function is very large, but it is not restricted with respect to the temporal derivative of the distribution function. The hydrodynamic equations corresponding to the lowest order of both expansions constitute two different hypersonic closures of the moment equations. For the subsequent orders in the expansions, closed sets of moment equations (hydrodynamic equations) are given. Special emphasis is made on the order of magnitude of the errors of the lowest-order hydrodynamic quantities. It is shown that if the heat flux vanishes initially, these errors are smaller than one might have expected from the ordinary scaling of the hypersonic closure. Also it is found that the normal solution of both expansions is a Gaussian distribution at the lowest order.
Quantum Fokker-Planck-Kramers equation and entropy production.
de Oliveira, Mário J
2016-07-01
We use a canonical quantization procedure to set up a quantum Fokker-Planck-Kramers equation that accounts for quantum dissipation in a thermal environment. The dissipation term is chosen to ensure that the thermodynamic equilibrium is described by the Gibbs state. An expression for the quantum entropy production that properly describes quantum systems in a nonequilibrium stationary state is also provided. The time-dependent solution is given for a quantum harmonic oscillator in contact with a heat bath. We also obtain the stationary solution for a system of two coupled harmonic oscillators in contact with reservoirs at distinct temperatures, from which we obtain the entropy production and the quantum thermal conductance. PMID:27575097
Solution of the Fokker-Planck equation in a wind turbine array boundary layer
NASA Astrophysics Data System (ADS)
Melius, Matthew S.; Tutkun, Murat; Cal, Raúl Bayoán
2014-07-01
Hot-wire velocity signals from a model wind turbine array boundary layer flow wind tunnel experiment are analyzed. In confirming Markovian properties, a description of the evolution of the probability density function of velocity increments via the Fokker-Planck equation is attained. Solution of the Fokker-Planck equation is possible due to the direct computation of the drift and diffusion coefficients from the experimental measurement data which were acquired within the turbine canopy. A good agreement is observed in the probability density functions between the experimental data and numerical solutions resulting from the Fokker-Planck equation, especially in the far-wake region. The results serve as a tool for improved estimation of wind velocity within the array and provide evidence that the evolution of such a complex and turbulent flow is also governed by a Fokker-Planck equation at certain scales.
Fokker-Planck equation with arbitrary dc and ac fields: continued fraction method.
Lee, Chee Kong; Gong, Jiangbin
2011-07-01
The continued fraction method (CFM) is used to solve the Fokker-Planck equation with arbitrary dc and ac fields. With an appropriate choice of basis functions, the Fokker-Planck equation is converted into a set of linear algebraic equations with short-ranged coupling and then CFM is implemented to obtain numerical solutions with high efficiency. Both a proposed perturbative CFM and the numerically exact matrix CFM are used to study the nonlinear response of driven systems, with their results compared to assess the validity regime of the perturbative approach. The proposed perturbative CFM approach needs scalar quantities only and hence is more efficient within its validity regime. Two nonlinear systems of different nature are used as examples: molecular dipole (rotational Brownian motion) and particle in a periodic potential (translational Brownian motion). The associated full dynamics is presented in the compact form of hysteresis loops. It is observed that as the strength of an AC driving field increases, pronounced nonlinear effects are manifested in the deformation of the hysteresis loops. PMID:21867110
Transport in the spatially tempered, fractional Fokker-Planck equation
Kullberg, A.; Del-Castillo-Negrete, Diego B
2012-01-01
A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, lambda, and on the order of the fractional derivative in space, alpha. An expansion of the TFD operator for large lambda is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit lambda -> infinity, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (lambda > 0) truncations, and alpha not equal 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit lambda -> infinity, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for alpha not equal 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any lambda > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with lambda, as J similar to lambda(-zeta), for alpha >= 1.75. However, for alpha <= 1.5, the steady-state current decays exponentially with lambda, as J similar to e(-xi lambda). In the presence of an asymmetry in the TFD operator, the tempering can lead
Transport in the spatially tempered, fractional Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Kullberg, A.; del-Castillo-Negrete, D.
2012-06-01
A study of truncated Lévy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, λ, and on the order of the fractional derivative in space, α. An expansion of the TFD operator for large λ is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit λ → ∞, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (λ > 0) truncations, and α ≠ 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit λ → ∞, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for α ≠ 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any λ > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with λ, as J ˜ λ-ζ, for α ⩾ 1.75. However, for α ⩽ 1.5, the steady-state current decays exponentially with λ, as J ˜ e-ξλ. In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the
NASA Astrophysics Data System (ADS)
Alrachid, Houssam; Lelièvre, Tony; Talhouk, Raafat
2016-05-01
We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non-linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.
Fokker-Planck equations for charged-particle transport in random fields.
NASA Technical Reports Server (NTRS)
Jokipii, J. R.
1972-01-01
The Fokker-Planck equations for charged-particle dynamics are rederived, extending somewhat the elegant discussion of Hasselmann and Wibberenz. It is shown that the usual results are obtae and the conclusions in many cases are correct over a very broad range in energy. In particular, the rate for pitch-angle scattering may be accurately given down to energies much lower than previously thought. Recent claims that these Fokker-Planck equations are in general incorrect are thus shown to be in error.
Fokker-Planck equation in the presence of a uniform magnetic field
NASA Astrophysics Data System (ADS)
Dong, Chao; Zhang, Wenlu; Li, Ding
2016-08-01
The Fokker-Planck equation in the presence of a uniform magnetic field is derived which has the same form as the case of no magnetic field but with different Fokker-Planck coefficients. The coefficients are calculated explicitly within the binary collision model, which are free from infinite sums of Bessel functions. They can be used to investigate relaxation and transport phenomena conveniently. The kinetic equation is also manipulated into the Landau form from which it is straightforward to compare with previous results and prove the conservation laws.
NASA Technical Reports Server (NTRS)
Jokipii, J. R.
1973-01-01
A derivation of the Fokker-Planck equation, based on the central limit theorem, is presented which clearly illustrates the conditions for its validity. It is reiterated that previous use of the Fokker-Planck equation in cosmic-ray transport is correct. Higher-order effects associated with magnetic mirroring and field line random walk at low energies are discussed heuristically.
The Fokker-Planck equation for the radiation transfer in a strongly magnetized plasma
NASA Astrophysics Data System (ADS)
Bonazzola, S.
1982-04-01
The Fokker-Planck equation for the radiation transfer in a strongly magnetized plasma is obtained by means of an approximation. It is noted that this equation requires less computer time than the Monte Carlo method and that it allows the effect of the induced processes to be taken into account without difficulty.
Conservative differencing of the electron Fokker-Planck transport equation
Langdon, A.B.
1981-01-12
We need to extend the applicability and improve the accuracy of kinetic electron transport codes. In this paper, special attention is given to modelling of e-e collisions, including the dominant contributions arising from anisotropy. The electric field and spatial gradient terms are also considered. I construct finite-difference analogues to the Fokker-Planck integral-differential collision operator, which conserve the particle number, momentum and energy integrals (sums) regardless of the coarseness of the velocity zoning. Such properties are usually desirable, but are especially useful, for example, when there are spatial regions and/or time intervals in which the plasma is cool, so that the collision operator acts rapidly and the velocity distribution is poorly resolved, yet it is crucial that gross conservation properties be respected in hydro-transport applications, such as in the LASNEX code. Some points are raised concerning spatial differencing and time integration.
Fokker-Planck-Kolmogorov Equation for fBm: Derivation and Analytical Solutions
NASA Astrophysics Data System (ADS)
Ünal, Gazanfer
2007-04-01
The Fokker-Planck-Kolmogorov (FPK) equation for Itô systems with fractional Brownian motion (fBm) has been derived. The generalized Liouville theorem has been proved. Analytic solutions to the FPK have been obtained via conserved quantities of the deterministic part.
Analytical and Numerical Solutions of Generalized Fokker-Planck Equations - Final Report
Prinja, Anil K.
2000-12-31
The overall goal of this project was to develop advanced theoretical and numerical techniques to quantitatively describe the spreading of a collimated beam of charged particles in space, in angle, and in energy, as a result of small deflection, small energy transfer Coulomb collisions with the target nuclei and electrons. Such beams arise in several applications of great interest in nuclear engineering, and include electron and ion radiotherapy, ion beam modification of materials, accelerator transmutation of waste, and accelerator production of tritium, to name some important candidates. These applications present unique and difficult modeling challenges, but from the outset are amenable to the language of ''transport theory'', which is very familiar to nuclear engineers and considerably less-so to physicists and material scientists. Thus, our approach has been to adopt a fundamental description based on transport equations, but the forward peakedness associated with charged particle interactions precludes a direct application of solution methods developed for neutral particle transport. Unique problem formulations and solution techniques are necessary to describe the transport and interaction of charged particles. In particular, we have developed the Generalized Fokker-Planck (GFP) approach to describe the angular and radial spreading of a collimated beam and a renormalized transport model to describe the energy-loss straggling of an initially monoenergetic distribution. Both analytic and numerical solutions have been investigated and in particular novel finite element numerical methods have been developed. In the first phase of the project, asymptotic methods were used to develop closed form solutions to the GFP equation for different orders of expansion, and was described in a previous progress report. In this final report we present a detailed description of (i) a novel energy straggling model based on a Fokker-Planck approximation but which is adapted for a
A covariant Fokker-Planck equation for a simple gas from relativistic kinetic theory
Chacon-Acosta, Guillermo; Dagdug, Leonardo; Morales-Tecotl, Hugo A.
2010-12-14
A manifestly covariant Fokker-Planck differential equation is derived for the case of a relativistic simple gas by taking a small momentum transfer approximation within the collision integral of the relativistic Boltzmann equation. We follow closely previous work, with the main difference that we keep manifest covariance at every stage of the analysis. In addition, we use the covariant Juettner distribution function to find a relativistic generalization of the Einstein's fluctuation-dissipation relation.
Moment-Preserving SN Discretizations for the One-Dimensional Fokker-Planck Equation
Warsa, James S.; Prinja, Anil K.
2012-06-14
The Fokker-Planck equation: (1) Describes the transport and interactions of charged particles, (2) Many small-angle scattering collisions, (3) Asymptotic limit of the Boltzmann equation (Pomraning, 1992), and (4) The Boltzmann collision operator becomes the angular Laplacian. SN angular discretization: (1) Angular flux is collocated at the SN quadrature points, (2) The second-order derivatives in the Laplacian term must be discretized, and (3) Weighted finite-difference method preserves zeroth and first moments (Morel, 1985). Moment-preserving methods: (1) Collocate the Fokker-Planck operator at the SN quadrature points, (2) Develop several related and/or equivalent methods, and (3) Motivated by discretizations for the angular derivative appearing in the transport equation in one-dimensional spherical coordinates.
Fractional Fokker-Planck Equation and Black-Scholes Formula in Composite-Diffusive Regime
NASA Astrophysics Data System (ADS)
Liang, Jin-Rong; Wang, Jun; Lǔ, Long-Jin; Gu, Hui; Qiu, Wei-Yuan; Ren, Fu-Yao
2012-01-01
In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α, H ( t)= X H ( S α ( t)), 0< α, H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equation and Black-Scholes formula. We obtain the fractional Fokker-Planck equation governing the dynamics of the probability density function of the composite-diffusive fractional Brownian motion and find the Black-Scholes differential equation driven by the stock asset X α, H ( t) and the corresponding Black-Scholes formula for the fair prices of European option.
Danos, Rebecca J.; Fiege, Jason D.; Shalchi, Andreas E-mail: fiege@physics.umanitoba.ca
2013-07-20
We present numerical solutions to both the standard and modified two-dimensional Fokker-Planck equations with adiabatic focusing and isotropic pitch-angle scattering. With the numerical solution of the particle distribution function, we then discuss the related numerical issues, calculate the parallel diffusion coefficient using several different methods, and compare our numerical solutions for the parallel diffusion coefficient to the analytical forms derived earlier. We find the numerical solution to the diffusion coefficient for both the standard and modified Fokker-Planck equations agrees with that of Shalchi for the mean squared displacement method of computing the diffusion coefficient. However, we also show the numerical solution agrees with that of Litvinenko and Shalchi and Danos when calculating the diffusion coefficient via the velocity correlation function.
Classical integrability for beta-ensembles and general Fokker-Planck equations
Rumanov, Igor
2015-01-15
Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here, we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g., there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system–a closed system of two nonlinear partial differential equations (PDEs) of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.
NASA Astrophysics Data System (ADS)
Sun, Yifei; Kumar, Mrinal
2015-05-01
In this paper, a tensor decomposition approach combined with Chebyshev spectral differentiation is presented to solve the high dimensional transient Fokker-Planck equations (FPE) arising in the simulation of polymeric fluids via multi-bead-spring (MBS) model. Generalizing the authors' previous work on the stationary FPE, the transient solution is obtained in a single CANDECOMP/PARAFAC decomposition (CPD) form for all times via the alternating least squares algorithm. This is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions, thereby decoupling it from them. As a result, the transient solution is obtained without resorting to expensive time stepping schemes. A new, relaxed approach for imposing the vanishing boundary conditions is proposed, improving the quality of the approximation. The asymptotic behavior of the temporal basis functions is studied. The proposed solver scales very well with the dimensionality of the MBS model. Numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix Riccati differential equation (for linear models) or Monte Carlo simulations (for nonlinear models).
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration. PMID:24329213
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
Bakhtiyari-Ramezani, M. Alinejad, N.; Mahmoodi, J.
2015-11-15
In the fusion devices, ions, H atoms, and H{sub 2} molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H{sub 2} molecules, and desorption of the recombined H{sub 2} molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.
Diffusion coefficients of Fokker-Planck equation for rotating dust grains in a fusion plasma
NASA Astrophysics Data System (ADS)
Bakhtiyari-Ramezani, M.; Mahmoodi, J.; Alinejad, N.
2015-11-01
In the fusion devices, ions, H atoms, and H2 molecules collide with dust grains and exert stochastic torques which lead to small variations in angular momentum of the grain. By considering adsorption of the colliding particles, thermal desorption of H atoms and normal H2 molecules, and desorption of the recombined H2 molecules from the surface of an oblate spheroidal grain, we obtain diffusion coefficients of the Fokker-Planck equation for the distribution function of fluctuating angular momentum. Torque coefficients corresponding to the recombination mechanism show that the nonspherical dust grains may rotate with a suprathermal angular velocity.
Solving the Fokker-Planck equation with the finite-element method
Galán, Roberto F.; Ermentrout, G. Bard; Urban, Nathaniel N.
2008-01-01
We apply an efficient approach from computational engineering, the finite-element method, to numerically solve the Fokker-Planck equation in two dimensions. This approach permits us to find the solution to stochastic problems that cannot be solved analytically. We illustrate our strategy with an example from neuroscience that recently has attracted considerable attention - synchronization of neural oscillators. In particular, we show that resonators (type II neural oscillators) respond and synchronize more reliably when provided correlated stochastic inputs than do integrators (type I neural oscillators). This result is consistent with recent experimental and computational work. We briefly discuss its relevance for neuroscience. PMID:18233721
Fokker Planck Rosenbluth-type equations for self-gravitating systems in the 1PN approximation
NASA Astrophysics Data System (ADS)
Ramos-Caro, Javier; González, Guillermo A.
2008-02-01
We present two formulations of Fokker Planck Rosenbluth-type (FPR) equations for many-particle self-gravitating systems, with first-order relativistic corrections in the post-Newtonian approach (1PN). The first starts from a covariant Fokker Planck equation for a simple gas, introduced recently by Chacón-Acosta and Kremer (2007 Phys. Rev. E 76 021201). The second derivation is based on the establishment of an 1PN-BBGKY hierarchy, developed systematically from the 1PN microscopic law of force and using the Klimontovich Dupree (KD) method. We close the hierarchy by the introduction of a two-point correlation function that describes adequately the relaxation process. This picture reveals an aspect that is not considered in the first formulation: the contribution of ternary correlation patterns to the diffusion coefficients, as a consequence of the nature of 1PN interaction. Both formulations can be considered as a generalization of the equation derived by Rezania and Sobouti (2000 Astron. Astrophys. 354 1110), to stellar systems where the relativistic effects of gravitation play a significant role.
Densmore, Jeffery D. Warsa, James S. Lowrie, Robert B. Morel, Jim E.
2009-09-01
The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.
Densmore, Jeffery D; Warsa, James S; Lowrie, Robert B; Morel, Jim E
2008-01-01
The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.
MODELING THE SUNSPOT NUMBER DISTRIBUTION WITH A FOKKER-PLANCK EQUATION
Noble, P. L.; Wheatland, M. S.
2011-05-01
Sunspot numbers exhibit large short-timescale (daily-monthly) variation in addition to longer-timescale variation due to solar cycles. A formal statistical framework is presented for estimating and forecasting randomness in sunspot numbers on top of deterministic (including chaotic) models for solar cycles. The Fokker-Planck approach formulated assumes a specified long-term or secular variation in sunspot number over an underlying solar cycle via a driver function. The model then describes the observed randomness in sunspot number on top of this driver function. We consider a simple harmonic choice for the driver function, but the approach is general and can easily be extended to include other drivers which account for underlying physical processes and/or empirical features of the sunspot numbers. The framework is consistent during both solar maximum and minimum, and requires no parameter restrictions to ensure non-negative sunspot numbers. Model parameters are estimated using statistically optimal techniques. The model agrees both qualitatively and quantitatively with monthly sunspot data even with the simplistic representation of the periodic solar cycle. This framework should be particularly useful for solar cycle forecasters and is complementary to existing modeling techniques. An analytic approximation for the Fokker-Planck equation is presented, which is analogous to the Euler approximation, which allows for efficient maximum likelihood estimation of large data sets and/or when using difficult to evaluate driver functions.
Yoon, E. S.; Chang, C. S.
2014-03-15
An approximate two-dimensional solver of the nonlinear Fokker-Planck-Landau collision operator has been developed using the assumption that the particle probability distribution function is independent of gyroangle in the limit of strong magnetic field. The isotropic one-dimensional scheme developed for nonlinear Fokker-Planck-Landau equation by Buet and Cordier [J. Comput. Phys. 179, 43 (2002)] and for linear Fokker-Planck-Landau equation by Chang and Cooper [J. Comput. Phys. 6, 1 (1970)] have been modified and extended to two-dimensional nonlinear equation. In addition, a method is suggested to apply the new velocity-grid based collision solver to Lagrangian particle-in-cell simulation by adjusting the weights of marker particles and is applied to a five dimensional particle-in-cell code to calculate the neoclassical ion thermal conductivity in a tokamak plasma. Error verifications show practical aspects of the present scheme for both grid-based and particle-based kinetic codes.
Analytical Solutions of the Fokker-Planck Equation for Generalized Morse and Hulthén Potentials
NASA Astrophysics Data System (ADS)
Anjos, R. C.; Freitas, G. B.; Coimbra-Araújo, C. H.
2016-01-01
In the present contribution we analytically calculate solutions of the transition probability of the Fokker-Planck equation (FPE) for both the generalized Morse potential and the Hulthén potential. The method is based on the formal analogy of the FPE with the Schrödinger equation using techniques from supersymmetric quantum mechanics.
Global well-posedness for the Fokker-Planck-Boltzmann equation in Besov-Chemin-Lerner type spaces
NASA Astrophysics Data System (ADS)
Liu, Zhengrong; Tang, Hao
2016-06-01
In this paper, motivated by [16], we use the Littlewood-Paley theory to establish some estimates on the nonlinear collision term, which enable us to investigate the Cauchy problem of the Fokker-Planck-Boltzmann equation. When the initial data is a small perturbation of the Maxwellian equilibrium state, under the Grad's angular cutoff assumption, the unique global solution for the hard potential case is obtained in the Besov-Chemin-Lerner type spaces C ([ 0 , ∞) ; L˜>ξ 2 (B2,rs)) with 1 ≤ r ≤ 2 and s > 3 / 2 or s = 3 / 2 and r = 1. Besides, we also obtain the uniform stability of the dependence on the initial data.
Deterministic proton transport solving a one dimensional Fokker-Planck equation
Marr, D.; Prael, R.; Adams, K.; Alcouffe, R.
1997-10-01
The transport of protons through matter is characterized by many interactions which cause small deflections and slight energy losses. The few which are catastrophic or cause large angle scattering can be viewed as extinction for many applications. The transport of protons at this level of approximation can be described by a Fokker Planck Equation. This equation is solved using a deterministic multigroup differencing scheme with a highly resolved set of discrete ordinates centered around the beam direction which is adequate to properly account for deflections and energy losses due to multiple Coulomb scattering. Comparisons with LAHET for a large variety of problems ranging from 800 MeV protons on a copper step wedge to 10 GeV protons on a sandwich of material are presented. The good agreement with the Monte Carlo code shows that the solution method is robust and useful for approximate solutions of selected proton transport problems.
Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
NASA Astrophysics Data System (ADS)
Ho, Choon-Lin
2011-04-01
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree ℓ = 1, 2, … , and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new Xℓ polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker-Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.
New Kinematic Model in comparing with Langevin equation and Fokker Planck Equation
NASA Astrophysics Data System (ADS)
Lee, Kyoung; Wang, Zhijian; Gardner, Robin
2010-03-01
An analytic approximate solution of New Kinematic Model with the boundary conditions is developed for the incompressible packing condition in Pebble Bed Reactors. It is based on velocity description of the packing density in the hopper. The packing structure can be presented with a jamming phenomenon from flow types. The gravity-driven macroscopic motions are governed not only by the geometry and external boundary conditions of silos and hoppers, but by flow prosperities of granular materials, such as friction, viscosity and porosity. The analytical formulas for the quasi-linear diffusion and convection coefficients of the velocity profile are obtained. Since it was found that the New Kinematic Model is dependent upon the granular packing density distribution, we are motivated to study the Langevin equation with friction under the influence of the Gravitational field. We also discuss the relation with the Fokker Planck Equation using Detailed balance and Metropolis-Hastings Algorithm. Markov chain Monte Carlo methods are shown to be a non-Maxwellian distribution function with the mean velocity of the field particles having an effective temperature.
Solution of the Fokker-Planck Equation with a Logarithmic Potential
NASA Astrophysics Data System (ADS)
Dechant, A.; Lutz, E.; Barkai, E.; Kessler, D. A.
2011-12-01
We investigate the diffusion of particles in an attractive one-dimensional potential that grows logarithmically for large | x| using the Fokker-Planck equation. An eigenfunction expansion shows that the Boltzmann equilibrium density does not fully describe the long-time limit of this problem. Instead this limit is characterized by an infinite covariant density. This non-normalizable density yields the mean square displacement of the particles, which for a certain range of parameters exhibits anomalous diffusion. In a symmetric potential with an asymmetric initial condition, the average position decays anomalously slowly. This problem also has applications outside the thermal context, as in the diffusion of the momenta of atoms in optical molasses.
NASA Astrophysics Data System (ADS)
Hager, Robert; Yoon, E. S.; Ku, S.; D'Azevedo, E. F.; Worley, P. H.; Chang, C. S.
2016-06-01
Fusion edge plasmas can be far from thermal equilibrium and require the use of a non-linear collision operator for accurate numerical simulations. In this article, the non-linear single-species Fokker-Planck-Landau collision operator developed by Yoon and Chang (2014) [9] is generalized to include multiple particle species. The finite volume discretization used in this work naturally yields exact conservation of mass, momentum, and energy. The implementation of this new non-linear Fokker-Planck-Landau operator in the gyrokinetic particle-in-cell codes XGC1 and XGCa is described and results of a verification study are discussed. Finally, the numerical techniques that make our non-linear collision operator viable on high-performance computing systems are described, including specialized load balancing algorithms and nested OpenMP parallelization. The collision operator's good weak and strong scaling behavior are shown.
Modelling of income distribution in the European Union with the Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Jagielski, Maciej; Kutner, Ryszard
2013-05-01
Herein, we applied statistical physics to study incomes of three (low-, medium- and high-income) society classes instead of the two (low- and medium-income) classes studied so far. In the frame of the threshold nonlinear Langevin dynamics and its threshold Fokker-Planck counterpart, we derived a unified formula for description of income of all society classes, by way of example, of those of the European Union in years 2006 and 2008. Hence, the formula is more general than the well known formula of Yakovenko et al.. That is, our formula well describes not only two regions but simultaneously the third region in the plot of the complementary cumulative distribution function vs. an annual household income. Furthermore, the known stylised facts concerning this income are well described by our formula. Namely, the formula provides the Boltzmann-Gibbs income distribution function for the low-income society class and the weak Pareto law for the medium-income society class, as expected. Importantly, it predicts (to satisfactory approximation) the Zipf law for the high-income society class. Moreover, the region of medium-income society class is now distinctly reduced because the bottom of high-income society class is distinctly lowered. This reduction made, in fact, the medium-income society class an intermediate-income society class.
NASA Astrophysics Data System (ADS)
Jang, Seogjoo
2016-06-01
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.
Jang, Seogjoo
2016-06-01
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath. PMID:27276940
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
NASA Astrophysics Data System (ADS)
Tanimura, Yoshitaka
2015-04-01
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as “Wigner space”), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.
Real-time and imaginary-time quantum hierarchal Fokker-Planck equations.
Tanimura, Yoshitaka
2015-04-14
We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material. PMID:25877565
NASA Astrophysics Data System (ADS)
Polson, James M.; Dunn, Taylor R.
2014-05-01
Brownian dynamics (BD) simulations are used to study the translocation dynamics of a coarse-grained polymer through a cylindrical nanopore. We consider the case of short polymers, with a polymer length, N, in the range N = 21-61. The rate of translocation is controlled by a tunable friction coefficient, γ0p, for monomers inside the nanopore. In the case of unforced translocation, the mean translocation time scales with polymer length as ⟨τ1⟩ ˜ (N - Np)α, where Np is the average number of monomers in the nanopore. The exponent approaches the value α = 2 when the pore friction is sufficiently high, in accord with the prediction for the case of the quasi-static regime where pore friction dominates. In the case of forced translocation, the polymer chain is stretched and compressed on the cis and trans sides, respectively, for low γ0p. However, the chain approaches conformational quasi-equilibrium for sufficiently large γ0p. In this limit the observed scaling of ⟨τ1⟩ with driving force and chain length supports the Fokker-Planck (FP) prediction that ⟨τ⟩ ∝ N/fd for sufficiently strong driving force. Monte Carlo simulations are used to calculate translocation free energy functions for the system. The free energies are used with the FP equation to calculate translocation time distributions. At sufficiently high γ0p, the predicted distributions are in excellent agreement with those calculated from the BD simulations. Thus, the FP equation provides a valid description of translocation dynamics for sufficiently high pore friction for the range of polymer lengths considered here. Increasing N will require a corresponding increase in pore friction to maintain the validity of the FP approach. Outside the regime of low N and high pore friction, the polymer is out of equilibrium, and the FP approach is not valid.
A data-driven alternative to the fractional Fokker-Planck equation
NASA Astrophysics Data System (ADS)
Pressé, Steve
2015-07-01
Anomalous diffusion processes are ubiquitous in biology and arise in the transport of proteins, vesicles and other particles. Such anomalously diffusive behavior is attributed to a number of factors within the cell including heterogeneous environments, active transport processes and local trapping/binding. There are a number of microscopic principles—such as power law jump size and/or waiting time distributions—from which the fractional Fokker-Planck equation (FFPE) can be derived and used to provide mechanistic insight into the origins of anomalous diffusion. On the other hand, it is fair to ask if other microscopic principles could also have given rise to the evolution of an observed density profile that appears to be well fit by an FFPE. Here we discuss another possible mechanistic alternative that can give rise to densities like those generated by FFPEs. Rather than to fit a density (or concentration profile) using a solution to the spatial FFPE, we reconstruct the profile generated by an FFPE using a regular FPE with a spatial and time-dependent force. We focus on the special case of the spatial FFPE for superdiffusive processes. This special case is relevant to, for example, active transport in a biological context. We devise a prescription for extracting such forces on synthetically generated data and provide an interpretation to the forces extracted. In particular, the time-dependence of forces could tell us about ATP depletion or changes in the cell's metabolic activity. Modeling anomalous behavior with normal diffusion driven by these effective forces yields an alternative mechanistic picture that, ultimately, could help motivate future experiments.
NASA Astrophysics Data System (ADS)
Taitano, W. T.; Chacón, L.; Simakov, A. N.; Molvig, K.
2015-09-01
In this study, we demonstrate a fully implicit algorithm for the multi-species, multidimensional Rosenbluth-Fokker-Planck equation which is exactly mass-, momentum-, and energy-conserving, and which preserves positivity. Unlike most earlier studies, we base our development on the Rosenbluth (rather than Landau) form of the Fokker-Planck collision operator, which reduces complexity while allowing for an optimal fully implicit treatment. Our discrete conservation strategy employs nonlinear constraints that force the continuum symmetries of the collision operator to be satisfied upon discretization. We converge the resulting nonlinear system iteratively using Jacobian-free Newton-Krylov methods, effectively preconditioned with multigrid methods for efficiency. Single- and multi-species numerical examples demonstrate the advertised accuracy properties of the scheme, and the superior algorithmic performance of our approach. In particular, the discretization approach is numerically shown to be second-order accurate in time and velocity space and to exhibit manifestly positive entropy production. That is, H-theorem behavior is indicated for all the examples we have tested. The solution approach is demonstrated to scale optimally with respect to grid refinement (with CPU time growing linearly with the number of mesh points), and timestep (showing very weak dependence of CPU time with time-step size). As a result, the proposed algorithm delivers several orders-of-magnitude speedup vs. explicit algorithms.
NASA Astrophysics Data System (ADS)
Lukassen, Laura; Oberlack, Martin
2014-11-01
As described in literature, non-Brownian particles in shear flow show a diffusive behavior due to hydrodynamic interactions. This shear-induced diffusion differs from the well-known Brownian diffusion, as there is no separation of time scales. That means that the configuration of non-Brownian particles changes on the same time scale as the hydrodynamic velocity. This fact impedes the derivation of a Fokker-Planck equation describing non-Brownian particles in pure position space. In this context, we derived a new Fokker-Planck approach in coupled position-velocity space to assure the validity of the Markov process assumption which is violated in pure position space formulation (Lukassen, Oberlack, Phys. Rev. E 89, 2014). Here, we present a further validation of our new Fokker-Planck approach that allows us to establish a relation to a modified purely position space Fokker-Planck equation. This backward transformation exhibits additional correction terms when compared to other position space Fokker-Planck equations in that context known from literature. Our extended approach shall enable a better stochastic description of non-Brownian particle flows. The work of L. Lukassen is supported by the ``Excellence Initiative'' of the German Federal and State Governments and the Graduate School of Computational Engineering at TU Darmstadt.
NASA Astrophysics Data System (ADS)
Palyulin, Vladimir V.; Chechkin, Aleksei V.; Metzler, Ralf
2014-11-01
Based on the space-fractional Fokker-Planck equation with a δ-sink term, we study the efficiency of random search processes based on Lévy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Lévy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Lévy flights, due to which Lévy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Lévy flights over the regular Brownian search. Contrary to the common belief that Lévy flights with a Lévy index α = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.
NASA Astrophysics Data System (ADS)
Yavorskij, V. A.; Andrushchenko, Zh. N.; Edenstrasser, J. W.; Goloborod'ko, V. Ya
1999-10-01
The five-dimensional (5D) drift kinetic Fokker-Planck equation for fast charged particles confined in a tokamak with a toroidal field (TF) ripple magnitude below the Goldston-White-Boozer stochasticity threshold is averaged over the banana and superbanana timescales. As a result, a three-dimensional (3D) Fokker-Planck equation in the constants of motion (COM) space describing the collisional transport of charged high-energy particles is obtained. Toroidally trapped particles with the toroidal precession being in resonance with the ripple perturbations are shown to yield the main contribution to the ripple induced transport. It is found that the rates of ripple superbanana diffusion and convection in the radial coordinate significantly exceed the corresponding rates of the bananas in the axisymmetric limit. The superbanana diffusion and convection shown to be dominant in the MeV energy range may be responsible for the loss of partially thermalized fusion products observed in the Tokamak fusion test reactor (TFTR) [S. J. Zweben, R. L. Boivin, C.-S. Chang et al., Nucl. Fusion 31, 2219 (1991); H. W. Herrmann, S. J. Zweben, D. S. Darrow et al., ibid. 37, 1437 (1997)].
Weibull Statistics for Upper Ocean Currents with the Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Chu, P. C.
2012-12-01
Upper oceans typically exhibit of a surface mixed layer with a thickness of a few to several hundred meters. This mixed layer is a key component in studies of climate, biological productivity and marine pollution. It is the link between the atmosphere and the deep ocean and directly affects the air-sea exchange of heat, momentum and gases. Vertically averaged horizontal currents across the mixed layer are driven by the residual between the Ekman transport and surface wind stress, and damped by the Rayleigh friction. A set of stochastic differential equations are derived for the two components of the current vector (u, v). The joint probability distribution function of (u, v) satisfies the Fokker-Planck equation (Chu, 2008, 2009), with the Weibull distribution as the solution for the current speed. To prove it, the PDF of the upper (0-50 m) tropical Pacific current speeds (w) was calculated from hourly ADCP data (1990-2007) at six stations for the Tropical Atmosphere Ocean project. In fact, it satisfies the two-parameter Weibull distribution reasonably well with different characteristics between El Nino and La Nina events: In the western Pacific, the PDF of w has a larger peakedness during the La Nina events than during the El Nino events; and vice versa in the eastern Pacific. However, the PDF of w for the lower layer (100-200 m) does not fit the Weibull distribution so well as the upper layer. This is due to the different stochastic differential equations between upper and lower layers in the tropical Pacific. For the upper layer, the stochastic differential equations, established on the base of the Ekman dynamics, have analytical solution, i.e., the Rayleigh distribution (simplest form of the Weibull distribution), for constant eddy viscosity K. Knowledge on PDF of w during the El Nino and La Nina events will improve the ensemble horizontal flux calculation, which contributes to the climate studies. Besides, the Weibull distribution is also identified from the
Asgarani, Somayeh
2015-02-01
A method of finding entropic form for a given stationary probability distribution and specified potential field is discussed, using the steady-state Fokker-Planck equation. As examples, starting with the Boltzmann and Tsallis distribution and knowing the force field, we obtain the Boltzmann-Gibbs and Tsallis entropies. Also, the associated entropy for the gamma probability distribution is found, which seems to be in the form of the gamma function. Moreover, the related Fokker-Planck equations are given for the Boltzmann, Tsallis, and gamma probability distributions. PMID:25768455
On the Derivation of a High-Velocity Tail from the Boltzmann-Fokker-Planck Equation for Shear Flow
NASA Astrophysics Data System (ADS)
Acedo, L.; Santos, A.; Bobylev, A. V.
2002-12-01
Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile U x ( y)= ay, where a is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function f( r, v)= f( V), with V≡ v- U( r), which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with a collision rate K( θ)∝lim ∈→0 ∈ -2 δ( θ- ∈), where θ is the scattering angle, in which case the nonlinear Boltzmann collision operator reduces to a Fokker-Planck operator. We have found analytically that for shear rates larger than a certain threshold value a th≃0.3520 ν (where ν is an average collision frequency and a th/ ν is the real root of the cubic equation 64 x 3+16 x 2+12 x-9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f( V; a)˜| V|-4- σ( a) Φ( ϕ; a), where ϕ≡tan V y / V x and the angular distribution function Φ( ϕ; a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition Φ( ϕ; a)= Φ( ϕ+ π; a) allows one to obtain the exponent σ( a) as a function of the shear rate. It diverges when a→ a th and tends to a minimum value σ min≃1.252 in the limit a→∞. As a consequence of this power-law decay for a> a th, all the velocity moments of a degree equal to or larger than 2+ σ( a) are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle ~ϕ( a), which rotates from ~ϕ=- π/4,3 π/4 when a→ a th to ~ϕ=0, π in the limit a→∞.
Numerical Methods for Nonlinear Fokker-Planck Collision Operator in TEMPEST
NASA Astrophysics Data System (ADS)
Kerbel, G.; Xiong, Z.
2006-10-01
Early implementations of Fokker-Planck collision operator and moment computations in TEMPEST used low order polynomial interpolation schemes to reuse conservative operators developed for speed/pitch-angle (v, θ) coordinates. When this approach proved to be too inaccurate we developed an alternative higher order interpolation scheme for the Rosenbluth potentials and a high order finite volume method in TEMPEST (,) coordinates. The collision operator is thus generated by using the expansion technique in (v, θ) coordinates for the diffusion coefficients only, and then the fluxes for the conservative differencing are computed directly in the TEMPEST (,) coordinates. Combined with a cut-cell treatment at the turning-point boundary, this new approach is shown to have much better accuracy and conservation properties.
NASA Astrophysics Data System (ADS)
Milovanov, Alexander V.
2001-04-01
The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 112 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given.
Milovanov, A V
2001-04-01
The formulation of the fractional Fokker-Planck-Kolmogorov (FPK) equation [Physica D 76, 110 (1994)] has led to important advances in the description of the stochastic dynamics of Hamiltonian systems. Here, the long-time behavior of the basic transport processes obeying the fractional FPK equation is analyzed. A derivation of the large-scale turbulent transport coefficient for a Hamiltonian system with 11 / 2 degrees of freedom is proposed in connection with the fractal structure of the particle chaotic trajectories. The principal transport regimes (i.e., a diffusion-type process, ballistic motion, subdiffusion in the limit of the frozen Hamiltonian, and behavior associated with self-organized criticality) are obtained as partial cases of the generalized transport law. A comparison with recent numerical and experimental studies is given. PMID:11308983
NASA Astrophysics Data System (ADS)
Herrmann, Michael; Niethammer, Barbara; Velázquez, Juan J. L.
2014-08-01
The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.
NASA Technical Reports Server (NTRS)
Ogallagher, J. J.; Maslyar, G. A., III
1975-01-01
A recently developed model predicts an energy dependent phase lag in the modulated cosmic ray density U(t) given by U(t) approximately equal to US (t - tau) where US is the solution to the Fokker-Planck equation under time independent conditions and tau is the average time spent by particles inside the modulating region. The delay times tau are functions of modulating parameters R (the radius of the modulating cavity), V (the solar wind velocity), and K (the effective average diffusion-coefficient which is a function of energy). This model is applied to predict the time evolution of the modulated cosmic ray proton spectrum over a simulated solar cycle. A modulation produced mostly by varying R over the solar cycle is less consistent with the observations.
Lo, Joseph; Shizgal, Bernie D
2006-11-21
Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered. PMID:17129090
NASA Astrophysics Data System (ADS)
He, S.; Ohara, N.
2015-12-01
Sub-grid spatial variability of snow still remains a challenge because of complicated snow accumulation and melt processes. In this study, the Fokker-Planck equation (FPE) was derived to simulate the probability distribution in two-dimensional probability space, snow depth versus snow density. This FPE describes the evolution of probability density function of the state variables throughout the snow season. The snow depth and snow density were selected as stochastic state variables while snow temperature was left deterministic. In this equation, the snowmelt and snow accumulation are treated as external sources of stochasticity. This means that both the mean and variance parts of these variables are taken into consideration in the FPE as advection convection and diffusion effects in the probability domain, respectively. The major challenge of the FPE is calculation of the covariance terms appeared in the diffusion terms due to limitation of the data. In this study, several possible simplifications of the FPE will be discussed. We will test the following hypotheses for the simplifications: (1) the convection correction term is small enough to be neglected compared to the mean convection term, (2) snowmelt and snow accumulation are independent each other, and (3) snow accumulation term can be evaluated by measured snow depth data. For the estimation of snowmelt terms, outputs of a distributed snow model may also be used to calculate the spatial and temporal distribution of snowmelt. Finally, the FPE will be solved with a numerical method in the probability domain of snow depth versus snow density.
NASA Astrophysics Data System (ADS)
Carlen, Eric A.; Maas, Jan
2014-11-01
Let denote the Clifford algebra over , which is the von Neumann algebra generated by n self-adjoint operators Q j , j = 1,…, n satisfying the canonical anticommutation relations, Q i Q j + Q j Q i = 2δ ij I, and let τ denote the normalized trace on . This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let denote the set of all positive operators such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space . The fermionic Fokker-Planck equation is a quantum-mechanical analog of the classical Fokker-Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker-Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.
Fokker-Planck/Transport model for neutral beam driven tokamaks
Killeen, J.; Mirin, A.A.; McCoy, M.G.
1980-01-01
The application of nonlinear Fokker-Planck models to the study of beam-driven plasmas is briefly reviewed. This evolution of models has led to a Fokker-Planck/Transport (FPT) model for neutral-beam-driven Tokamaks, which is described in detail. The FPT code has been applied to the PLT, PDX, and TFTR Tokamaks, and some representative results are presented.
Effects of the Tempered Aging and the Corresponding Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Deng, Weihua; Wang, Wanli; Tian, Xinchun; Wu, Yujiang
2016-07-01
In the renewal processes, if the waiting time probability density function is a tempered power-law distribution, then the process displays a transition dynamics; and the transition time depends on the parameter λ of the exponential cutoff. In this paper, we discuss the aging effects of the renewal process with the tempered power-law waiting time distribution. By using the aging renewal theory, the p-th moment of the number of renewal events n_a(t_a, t) in the interval (t_a, t_a+t) is obtained for both the weakly and strongly aged systems; and the corresponding surviving probabilities are also investigated. We then further analyze the tempered aging continuous time random walk and its Einstein relation, and the mean square displacement is attained. Moreover, the tempered aging diffusion equation is derived.
Fokker-Planck formalism in magnetic resonance simulations.
Kuprov, Ilya
2016-09-01
This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use. PMID:27470597
Fokker-Planck formalism in magnetic resonance simulations
NASA Astrophysics Data System (ADS)
Kuprov, Ilya
2016-09-01
This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.
Fokker-Planck model of hydrodynamics.
Singh, S K; Ansumali, Santosh
2015-03-01
We present a phenomenological description of the hydrodynamics in terms of the Fokker-Planck (FP) equation for one-particle distribution function. Similar to the Boltzmann equation or the Bhatnager-Gross-Krook (BGK) model, this approach is thermodynamically consistent and has the H theorem. In this model, transport coefficients as well as the equation of state can be provided independently. This approach can be used as an alternate to BGK-based methods as well as the direct simulation Monte Carlo method for the gaseous flows. PMID:25871242
Sliusarenko, O. Yu.; Chechkin, A. V.; Slyusarenko, Yu. V.
2015-04-15
By generalizing Bogolyubov’s reduced description method, we suggest a formalism to derive kinetic equations for many-body dissipative systems in external stochastic field. As a starting point, we use a stochastic Liouville equation obtained from Hamilton’s equations taking dissipation and stochastic perturbations into account. The Liouville equation is then averaged over realizations of the stochastic field by an extension of the Furutsu-Novikov formula to the case of a non-Gaussian field. As the result, a generalization of the classical Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is derived. In order to get a kinetic equation for the single-particle distribution function, we use a regular cutoff procedure of the BBGKY hierarchy by assuming weak interaction between the particles and weak intensity of the field. Within this approximation, we get the corresponding Fokker-Planck equation for the system in a non-Gaussian stochastic field. Two particular cases are discussed by assuming either Gaussian statistics of external perturbation or homogeneity of the system.
ICPP: Numerical Fokker-Planck calculations in nonuniform grids
NASA Astrophysics Data System (ADS)
Bizarro, João P. S.
2000-10-01
The Fokker-Planck equation arises in a wide class of problems in plasma physics, so numerical schemes that provide efficient, accurate, and stable solutions to that equation are always welcome. One way to accomplish this is via nonuniform grids, which allow the use of different mesh sizes according to the real needs of the physical problem under consideration. The extension of the standard finite-difference approach to general nonuniform grids, taking into account proper weighting coefficients, has already been presented, and the results have been rather conclusive [J. P. S. Bizarro and P. Rodrigues, Nucl. Fusion Vol. 37, 1509 (1997)]. Besides reviewing what has been achieved with nonuniform grids, a numerical scheme that is accurate to second order (both in time step and mesh size) is here extended and detailed. Such an analysis is rigourous for one-dimensional Fokker-Planck equations, and is generalized to two-dimensional equations. The constraints on the design of the nonuniform grid are discussed, as well as the particle and energy conservation properties. The conditions under which the nonuniformity correction in the weighting coefficients is essential to secure physically meaningful solutions are also analyzed. The proposed scheme is shown to efficiently handle both linear and weakly nonlinear problems and, in addition, its ability to provide solutions to stronger nonlinear situations is demonstrated. Some particular problems in the field of plasma physics (e.g., Coulomb collisions, Compton scattering by an electronic population, and the rf heating and current drive of thermonuclear reactors) are solved in order to illustrate several features, most particularly the usefulness of nonuniform grids in reducing computational effort and in increasing accuracy.
Bounce-averaged Fokker-Planck code for stellarator transport
Mynick, H.E.; Hitchon, W.N.G.
1985-07-01
A computer code for solving the bounce-averaged Fokker-Planck equation appropriate to stellarator transport has been developed, and its first applications made. The code is much faster than the bounce-averaged Monte-Carlo codes, which up to now have provided the most efficient numerical means for studying stellarator transport. Moreover, because the connection to analytic kinetic theory of the Fokker-Planck approach is more direct than for the Monte-Carlo approach, a comparison of theory and numerical experiment is now possible at a considerably more detailed level than previously.
NASA Astrophysics Data System (ADS)
Karpov, S. A.; Potapenko, I. F.
2015-10-01
A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.
Karpov, S. A.; Potapenko, I. F.
2015-10-15
A stochastic method of simulation of Coulomb interaction is considered. The main idea of the method is to approximate the nonlinear Landau kinetic collision integral by the Boltzmann integral. In its realization, the method can be attributed to a wide class of Monte Carlo-type methods. It is easily combined with the existing particle methods used to simulate collisionless plasmas. This is important for simulation of the dynamics of both laboratory and space plasmas when the mean free path of plasma particles is comparable with the plasma inhomogeneity scale length. Illustrative examples of relaxation of two-temperature plasma being subject to a high-frequency alternating electric field are given, and differences from their classical description are considered. The method satisfies the conservation laws for the number of particles, momentum, and energy and is simple and efficient in implementation.
NASA Astrophysics Data System (ADS)
Bianucci, Marco
2015-05-01
In this paper using a projection approach and defining the adjoint-Lie time evolution of differential operators, that generalizes the ordinary time evolution of functions, we obtain a Fokker-Planck equation for the distribution function of a part of interest of a large class of dynamical systems. The main assumptions are the weak interaction between the part of interest and the rest of the system (typically non linear) and the average linear response to external perturbations of the irrelevant part. We do not use ad hoc statistical assumptions to introduce as given a priori phenomenological equilibrium or transport coefficients. The drift terms induced by the interaction with the irrelevant part is obtained with a procedure that is reminiscent of that developed some years ago by Bianucci and Grigolini (see for example (Bianucci et al 1995 Phys. Rev. E 51 3002)) to derive in a ‘genuine’ way thermodynamics and statistical mechanics of macroscopic variables of interest starting from microscopic dynamics. However here we stay in a more broad and formal context where the system of interest could be dissipative and the interaction between the two systems could be non Hamiltonian, thus the approach of the cited paper can not be used to obtain the diffusion part of the Fokker-Planck equation. To face the problem of dealing with the series of differential operators stemming from the projection approach applied to this general case, we introduce the formalism of the Lie derivative and the corresponding adjoint-Lie time evolution of differential operators. In this theoretical framework we are able to obtain well defined analytic functions both for the drift and the diffusion coefficients of the Fokker-Planck equation. We think that the basic elements of Lie algebra introduced in our projection approach can be useful to achieve even more general and more formally elegant results than those here presented. Thus we consider this paper as a first step of this formal path to
Prinja, A.K.
1995-08-01
We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S{sub N} angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes.
Fokker Planck theory for energetic electron deposition in laser fusion
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2014-10-01
We have developed a Fokker Planck model to calculate the transport and deposition of energetic electrons, produced for instance by the two plasmon decay instability at the quarter critical surface. In steady state, the Fokker Planck equation reduces to a single universal equation in energy and space, an equation which appears to be quite simple, but which has a rather unconventional boundary condition. The equation is equally valid in planar and spherical geometry, and it depends on only a single parameter, the charge state Z. Hence one can solve for a universal solution, valid for each Z. An asymptotic solution to this equation will be presented, which allows the heating of the main plasma to be calculated from a simple analytical expression. A more accurate solution in terms of a Bessel function expansion will also be presented. From this, one obtains a heating rate which can be simply incorporated into fluid simulations.
Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model
NASA Astrophysics Data System (ADS)
Wu, Kung-Chien
2015-09-01
In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543-1608, 2004) for Boltzmann equation, but the Fokker-Planck term in this paper creates some technical difficulties.
NASA Technical Reports Server (NTRS)
Englert, G. W.
1971-01-01
A model of the random walk is formulated to allow a simple computing procedure to replace the difficult problem of solution of the Fokker-Planck equation. The step sizes and probabilities of taking steps in the various directions are expressed in terms of Fokker-Planck coefficients. Application is made to many particle systems with Coulomb interactions. The relaxation of a highly peaked velocity distribution of particles to equilibrium conditions is illustrated.
NASA Astrophysics Data System (ADS)
Taitano, W. T.; Chacón, L.; Simakov, A. N.
2016-08-01
In this study, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth-Fokker-Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, vth (t), and explicitly considers the resulting inertial terms in the Fokker-Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker-Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic vth-ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. In particular, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.
Fokker-Planck modeling of current penetration during electron cyclotron current drive
Merkulov, A.; Westerhof, E.; Schueller, F. C.
2007-05-15
The current penetration during electron cyclotron current drive (ECCD) on the resistive time scale is studied with a Fokker-Planck simulation, which includes a model for the magnetic diffusion that determines the parallel electric field evolution. The existence of the synergy between the inductive electric field and EC driven current complicates the process of the current penetration and invalidates the standard method of calculation in which Ohm's law is simply approximated by j-j{sub cd}={sigma}E. Here it is proposed to obtain at every time step a self-consistent approximation to the plasma resistivity from the Fokker-Planck code, which is then used in a concurrent calculation of the magnetic diffusion equation in order to obtain the inductive electric field at the next time step. A series of Fokker-Planck calculations including a self-consistent evolution of the inductive electric field has been performed. Both the ECCD power and the electron density have been varied, thus varying the well known nonlinearity parameter for ECCD P{sub rf}[MW/m{sup -3}]/n{sub e}{sup 2}[10{sup 19} m{sup -3}] [R. W. Harvey et al., Phys. Rev. Lett 62, 426 (1989)]. This parameter turns out also to be a good predictor of the synergetic effects. The results are then compared with the standard method of calculations of the current penetration using a transport code. At low values of the Harvey parameter, the standard method is in quantitative agreement with Fokker-Planck calculations. However, at high values of the Harvey parameter, synergy between ECCD and E{sub parallel} is found. In the case of cocurrent drive, this synergy leads to the generation of large amounts of nonthermal electrons and a concomitant increase of the electrical conductivity and current penetration time. In the case of countercurrent drive, the ECCD efficiency is suppressed by the synergy with E{sub parallel} while only a small amount of nonthermal electrons is produced.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
Morel, J.E.
1987-01-01
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs.
Dimensional interpolation for nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Fred
2005-09-01
Dimensional interpolation has been used successfully by physicists and chemists to solve the Schroedinger equation for atoms and complex molecules. The same basic idea can be used to solve the Fokker-Planck equation for nonlinear filters. In particular, it is well known (by physicists) that two Schroedinger equations are equivalent to two Fokker-Planck equations. Moreover, we can avoid the Schroedinger equation altogether and use dimensional interpolation directly on the Fokker-Planck equation. Dimensional interpolation sounds like a crazy idea, but it works. We will attempt to make this paper accessible to normal engineers who do not have quantum mechanics for breakfast.
Fokker-Planck description of wealth dynamics and the origin of Pareto's law
NASA Astrophysics Data System (ADS)
Boghosian, Bruce
2014-05-01
The so-called "Yard-Sale Model" of wealth distribution posits that wealth is transferred between economic agents as a result of transactions whose size is proportional to the wealth of the less wealthy agent. In recent work [B. M. Boghosian, Phys. Rev. E89, 042804 (2014)], it was shown that this results in a Fokker-Planck equation governing the distribution of wealth. With the addition of a mechanism for wealth redistribution, it was further shown that this model results in stationary wealth distributions that are very similar in form to Pareto's well-known law. In this paper, a much simpler derivation of that Fokker-Planck equation is presented.
Fokker-Planck approach to the pulse packet propagation in synfire chain.
Câteau, H; Fukai, T
2001-01-01
We applied the Fokker-Planck method to the so-called 'synfire chain' network model and showed how a synchronous population spike (pulse packet) evolves to a narrow pulse packet (width < 1 ms) or fades away, depending on its initial size and width. The results of numerical integration of the Fokker-Planck equation are in good agreement with those of simulations on a network of leaky integrate-and-fire neurons. For a narrow input pulse packet, the integration of the Fokker-Planck equation requires careful numerical treatment. However, we can construct a precise analytical waveform of an output packet, which proves valid for narrow input pulse packets, from the stationary solution to the Fokker-Planck equation and a previously proposed approximate input-output relationship. Our methods enable us also to understand an essential role of the synaptic noise in the evolution, the peculiar temporal evolution of a broader pulse packets, and the irrelevance of the refractory period in determining the waveform of a pulse packet. Furthermore, we elucidate possible functional roles of multiple interactive pulse packets in spatiotemporal information processing, i.e. the association of information and the temporal competition. PMID:11665762
Two temperature gas equilibration model with a Fokker-Planck type collision operator
NASA Astrophysics Data System (ADS)
Méndez, A. R.; Chacón-Acosta, G.; García-Perciante, A. L.
2014-01-01
The equilibration process of a binary mixture of gases with two different temperatures is revisited using a Fokker-Planck type equation. The collision integral term of the Boltzmann equation is approximated by a Fokker-Planck differential collision operator by assuming that one of the constituents can be considered as a background gas in equilibrium while the other species diffuses through it. As a main result the coefficients of the linear term and of the first derivative are modified by the temperature and kinetic energy difference of the two species. These modifications are expected to influence the form of the solution for the distribution function and the corresponding transport equations. When temperatures are equal, the usual result of a Rayleigh gas is recovered.
Simulating transient dynamics of the time-dependent time fractional Fokker-Planck systems
NASA Astrophysics Data System (ADS)
Kang, Yan-Mei
2016-09-01
For a physically realistic type of time-dependent time fractional Fokker-Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker-Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.
Fokker Planck and Krook theory for energetic electron deposition in laser fusion
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2015-11-01
We have developed a Fokker Planck and Krook model to calculate the transport and deposition of energetic electrons, produced for instance by the two plasmon decay instability at the quarter critical surface of a laser produced plasma. In steady state, the Fokker Planck equation reduces to a single universal equation in energy and space, an equation which whose asymptotic solution we calculate. The Krook theory also gives rise to an analytic expression solution. From each, one can calculate the spatially dependent heating of the interior plasma, which can be implemented at each time step in a fluid simulation. The equation is equally valid in planar and spherical geometry, and it depends on only a single parameter, the charge state Z. Hence one can solve for a universal solution, valid for each Z. the two approaches will be compared and discussed. We look to cooperate with anyone having a more advanced simulation capability, Direct Simulation Monte Carlo or Fokker Planck, who is willing to test our results. Work supported by the NRL Laser fusion program, DOE- NNSA and ONR.
Current dependence of spin torque switching rate based on Fokker-Planck approach
Taniguchi, Tomohiro Imamura, Hiroshi
2014-05-07
The spin torque switching rate of an in-plane magnetized system in the presence of an applied field is derived by solving the Fokker-Planck equation. It is found that three scaling currents are necessary to describe the current dependence of the switching rate in the low-current limit. The dependences of these scaling currents on the applied field strength are also studied.
NASA Astrophysics Data System (ADS)
Frank, T. D.
2008-02-01
We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.
Fokker Planck and Krook theory of energetic electron transport in a laser produced plasma
NASA Astrophysics Data System (ADS)
Manheimer, Wallace; Colombant, Denis
2015-09-01
Various laser plasma instabilities, such as the two plasma decay instability and the stimulated Raman scatter instability, produce large quantities of energetic electrons. How these electrons are transported and heat the plasma are crucial questions for laser fusion. This paper works out a Fokker Planck and Krook theory for such transport and heating. The result is a set of equations, for which one can find a simple asymptotic approximation for the solution, for the Fokker Planck case, and an exact solution for the Krook case. These solutions are evaluated and compared with one another. They give rise to expressions for the spatially dependent heating of the background plasma, as a function of the instantaneous laser and plasma parameters, in either planar or spherical geometry. These formulas are simple, universal (depending weakly only on the single parameter Z, the charge state), and can be easily be incorporated into a fluid simulation.
Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
NASA Astrophysics Data System (ADS)
Heinsalu, E.; Patriarca, M.; Goychuk, I.; Hänggi, P.
2007-09-01
We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.
The Fokker-Planck law of diffusion and pattern formation in heterogeneous environments.
Bengfort, Michael; Malchow, Horst; Hilker, Frank M
2016-09-01
We analyze the influence of spatially inhomogeneous diffusion on several common ecological problems. Diffusion is modeled with Fick's law and the Fokker-Planck law of diffusion. We discuss the differences between the two formalisms and when to use either the one or the other. In doing so, we start with a pure diffusion equation, then turn to a reaction-diffusion system with one logistically growing component which invades the spatial domain. We also look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator-prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in case of Fokker-Planck diffusion. PMID:26803768
Fokker Planck and Krook theory of energetic electron transport in a laser produced plasma
Manheimer, Wallace; Colombant, Denis
2015-09-15
Various laser plasma instabilities, such as the two plasma decay instability and the stimulated Raman scatter instability, produce large quantities of energetic electrons. How these electrons are transported and heat the plasma are crucial questions for laser fusion. This paper works out a Fokker Planck and Krook theory for such transport and heating. The result is a set of equations, for which one can find a simple asymptotic approximation for the solution, for the Fokker Planck case, and an exact solution for the Krook case. These solutions are evaluated and compared with one another. They give rise to expressions for the spatially dependent heating of the background plasma, as a function of the instantaneous laser and plasma parameters, in either planar or spherical geometry. These formulas are simple, universal (depending weakly only on the single parameter Z, the charge state), and can be easily be incorporated into a fluid simulation.
Fokker-Planck-DSMC algorithm for simulations of rarefied gas flows
NASA Astrophysics Data System (ADS)
Gorji, M. Hossein; Jenny, Patrick
2015-04-01
A Fokker-Planck based particle Monte Carlo algorithm was devised recently for simulations of rarefied gas flows by the authors [1-3]. The main motivation behind the Fokker-Planck (FP) model is computational efficiency, which could be gained due to the fact that the resulting stochastic processes are continuous in velocity space. This property of the model leads to simulations where the computational cost becomes independent of the Knudsen number (Kn) [3]. However, the Fokker-Planck model which can be seen as a diffusion approximation of the Boltzmann equation, becomes less accurate as Kn increases. In this study we propose a hybrid Fokker-Planck-Direct Simulation Monte Carlo (FP-DSMC) solution method, which is applicable for the whole range of Kn. The objective of this algorithm is to retain the efficiency of the FP scheme at low Kn (Kn ≪ 1) and to employ conventional DSMC at high Kn (Kn ≫ 1). Since the computational particles employed by the FP model represent the same data as in DSMC, the coupling between the two methods is straightforward. The new ingredient is a switching criterion which would ideally result in a hybrid scheme with the efficiency of the FP method and the accuracy of DSMC for the whole Kn-range. Here, we adopt the number of collisions in a given computational cell and for a given time step size as a decision criterion in order to switch between the FP model and DSMC. For assessment of the hybrid algorithm, different test cases including flow impingement and flow expansion through a slit were studied. Both accuracy and efficiency of the model are shown to be excellent for the presented test cases.
NASA Astrophysics Data System (ADS)
Tang, Xian-Zhu; Berk, H. L.; Guo, Zehua; McDevitt, C. J.
2014-03-01
Across a transition layer of disparate plasma temperatures, the high energy tail of the plasma distribution can have appreciable deviations from the local Maxwellian distribution due to the Knudson layer effect. The Fokker-Planck equation for the tail particle population can be simplified in a series of practically useful limiting cases. The first is the approximation of background Maxwellian distribution for linearizing the collision operator. The second is the supra-thermal particle speed ordering of vTi ≪ v ≪ vTe for the tail ions and vTi ≪ vTe ≪ v for the tail electrons. Keeping both the collisional drag and energy scattering is essential for the collision operator to produce a Maxwellian tail distribution. The Fokker-Planck model for following the tail ion distribution for a given background plasma profile is explicitly worked out for systems of one spatial dimension, in both slab and spherical geometry. A third simplification is an expansion of the tail particle distribution using the spherical harmonics, which are eigenfunctions of the pitch angle scattering operator. This produces a set of coupled Fokker-Planck equations that contain energy-dependent spatial diffusion terms in two coordinates (position and energy), which originate from pitch angle scattering in the original Fokker-Planck equation. It is shown that the well-known diffusive Fokker-Planck model is a poor approximation of the two-mode truncation model, which itself has fundamental deficiency compared with the three-mode truncation model. The cause is the lack of even-symmetry representation in pitch dependence in the two-mode truncation model.
Fokker-Planck description for the queue dynamics of large tick stocks
NASA Astrophysics Data System (ADS)
Garèche, A.; Disdier, G.; Kockelkoren, J.; Bouchaud, J.-P.
2013-09-01
Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. “Jump” events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.
Vlasov-Fokker-Planck Simulation of a Collisional Ion-Electron Shockwave
NASA Astrophysics Data System (ADS)
Taitano, William; Knoll, Dana; Prinja, Anil
2012-10-01
There has been recent increased interest in a range of kinetic plasma physics phenomena which may be important in simulating ICF pellet performance. [1] have numerically demonstrated the limitations of the classic Spitzer, Braginski fluid closures in collisional plasmas for shockwave problems. [1] has shown the importance of modeling kinetic effects for scale lengths of shockwave much larger than the ion collision mean free path. In [1], the ions were modeled kinetically using the Fokker-Planck approximation while the electrons were modeled as a fluid. An investigation of a full kinetic treatment of electron with collision is computationally intractable with standard explicit schemes due to collision CFL limitation that requires resolving the electron-electron collision timescale. [2] has developed a new, fully implicit and discretely consistent moment based accelerator method to solve the full ion-electron kinetic Vlasov-Ampere system. A similar moment based accelerator will be extended to a collisionless shock problem in order to accelerate the Fokker-Planck collision source in the kinetic equations. In the presentation, we provide some preliminary results. [4pt] [1] M. Casanova and O. Larroche, Phys. Rev. Let. 67-(16), 1991. [0pt] [2] W.T. Taitano et al. SISC in review.
A fractional Fokker-Planck model for anomalous diffusion
Anderson, Johan; Kim, Eun-jin; Moradi, Sara
2014-12-15
In this paper, we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality of the stable Lévy distribution. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior.
Mapping the Monte Carlo scheme to Langevin dynamics: a Fokker-Planck approach.
Cheng, X Z; Jalil, M B A; Lee, Hwee Kuan; Okabe, Yutaka
2006-02-17
We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors alpha, unlike previous time-quantified MC methods which break down at low alpha, where precessional motion dominates. PMID:16606044
Mapping the Monte Carlo Scheme to Langevin Dynamics: A Fokker-Planck Approach
NASA Astrophysics Data System (ADS)
Cheng, X. Z.; Jalil, M. B.; Lee, Hwee Kuan; Okabe, Yutaka
2006-02-01
We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and diffusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time-quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also find that our Metropolis MC method is accurate for a large range of damping factors α, unlike previous time-quantified MC methods which break down at low α, where precessional motion dominates.
NASA Technical Reports Server (NTRS)
Hizanidis, Kyriakos
1989-01-01
The relativistic motion of electrons in an intense electromagnetic wave packet propagating obliquely to a uniform magnetic field is analytically studied on the basis of the Fokker-Planck-Kolmogorov (FPK) approach. The wavepacket consists of circularly polarized electron-cyclotron waves. The dynamical system in question is shown to be reducible to one with three degrees of freedom. Within the framework of the Hamiltonian analysis the nonlinear diffusion tensor is derived, and it is shown that this tensor can be separated into zeroth-, first-, and second-order parts with respect to the relative bandwidth. The zeroth-order part describes diffusive acceleration along lines of constant unperturbed Hamiltonian. The second-order part, which corresponds to the longest time scale, describes diffusion across those lines. A possible transport theory is outlined on the basis of this separation of the time scales.
A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas
Xiong, Z; Cohen, R H; Rognlien, T D; Xu, X Q
2007-04-18
A high-order finite volume algorithm is developed for the Fokker-Planck Operator (FPO) describing Coulomb collisions in strongly magnetized plasmas. The algorithm is based on a general fourth-order reconstruction scheme for an unstructured grid in the velocity space spanned by parallel velocity and magnetic moment. The method provides density conservation and high-order-accurate evaluation of the FPO independent of the choice of the velocity coordinates. As an example, a linearized FPO in constant-of-motion coordinates, i.e. the total energy and the magnetic moment, is developed using the present algorithm combined with a cut-cell merging procedure. Numerical tests include the Spitzer thermalization problem and the return to isotropy for distributions initialized with velocity space loss cones. Utilization of the method for a nonlinear FPO is straightforward but requires evaluation of the Rosenbluth potentials.
Nonlinear Kramers equation associated with nonextensive statistical mechanics.
Mendes, G A; Ribeiro, M S; Mendes, R S; Lenzi, E K; Nobre, F D
2015-05-01
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results. PMID:26066118
High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications
Duclous, Roland Dubroca, Bruno Filbet, Francis Tikhonchuk, Vladimir
2009-08-01
A high order, deterministic direct numerical method is proposed for the non-relativistic 2D{sub x}x3D{sub v} Vlasov-Maxwell system, coupled with Fokker-Planck-Landau collision operators. The magnetic field is perpendicular to the 2D{sub x} plane surface of computation, whereas the electric fields occur in this plane. Such a system is devoted to modelling of electron transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It is able to describe the kinetics of the plasma electrons in the nonlocal equilibrium regime, and permits to consider a large anisotropy degree of the distribution function. We develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Fast algorithms are employed, which makes this direct approach computationally affordable for simulations of hundreds of collisional times.
A Fokker-Planck study of the eigenmodes in an unmagnetized pair plasma
Zhao Bin; Zheng Jian
2007-06-15
Linearized Fokker-Planck equations for unmagnetized pair plasmas are solved as an eigenvalue problem to investigate the sound waves and Langmuir waves. The frequencies and damping rates of sound waves and Langmuir waves as a function of k{lambda} and k{lambda}{sub D} are presented, where k is the wave number, {lambda} is the mean-free path, and {lambda}{sub D} is the Debye length. It is found that no electrostatic field is evolved in the process of sound wave. As a consequence, Landau damping is not relevant to the sound waves in a pair plasma. The damping mechanics of sound waves is fully governed by the Coulomb collisions. The valid regimes of fluid descriptions for the waves are also discussed by comparing with the computational results.
Compact Collision Kernels for Hard Sphere and Coulomb Cross Sections; Fokker-Planck Coefficients
Chang Yongbin; Shizgal, Bernie D.
2008-12-31
A compact collision kernel is derived for both hard sphere and Coulomb cross sections. The difference between hard sphere interaction and Coulomb interaction is characterized by a parameter {eta}. With this compact collision kernel, the calculation of Fokker-Planck coefficients can be done for both the Coulomb and hard sphere interactions. The results for arbitrary order Fokker-Planck coefficients are greatly simplified. An alternate form for the Coulomb logarithm is derived with concern to the temperature relaxation in a binary plasma.
Full linearized Fokker-Planck collisions in neoclassical transport simulations
NASA Astrophysics Data System (ADS)
Belli, E. A.; Candy, J.
2012-01-01
The complete linearized Fokker-Planck collision operator has been implemented in the drift-kinetic code NEO (Belli and Candy 2008 Plasma Phys. Control. Fusion 50 095010) for the calculation of neoclassical transport coefficients and flows. A key aspect of this work is the development of a fast numerical algorithm for treatment of the field particle operator. This Eulerian algorithm can accurately treat the disparate velocity scales that arise in the case of multi-species plasmas. Specifically, a Legendre series expansion in ξ (the cosine of the pitch angle) is combined with a novel Laguerre spectral method in energy to ameliorate the rapid numerical precision loss that occurs for traditional Laguerre spectral methods. We demonstrate the superiority of this approach to alternative spectral and finite-element schemes. The physical accuracy and limitations of more commonly used model collision operators, such as the Connor and Hirshman-Sigmar operators, are studied, and the effects on neoclassical impurity poloidal flows and neoclassical transport for experimental parameters are explored.
Híjar, Humberto
2015-02-01
We study the Brownian motion of a particle bound by a harmonic potential and immersed in a fluid with a uniform shear flow. We describe this problem first in terms of a linear Fokker-Planck equation which is solved to obtain the probability distribution function for finding the particle in a volume element of its associated phase space. We find the explicit form of this distribution in the stationary limit and use this result to show that both the equipartition law and the equation of state of the trapped particle are modified from their equilibrium form by terms increasing as the square of the imposed shear rate. Subsequently, we propose an alternative description of this problem in terms of a generalized Langevin equation that takes into account the effects of hydrodynamic correlations and sound propagation on the dynamics of the trapped particle. We show that these effects produce significant changes, manifested as long-time tails and resonant peaks, in the equilibrium and nonequilibrium correlation functions for the velocity of the Brownian particle. We implement numerical simulations based on molecular dynamics and multiparticle collision dynamics, and observe a very good quantitative agreement between the predictions of the model and the numerical results, thus suggesting that this kind of numerical simulations could be used as complement of current experimental techniques. PMID:25768490
Anomalous rotational relaxation: a fractional Fokker-Planck equation approach.
Aydiner, Ekrem
2005-04-01
In this study we have analytically obtained the relaxation function in terms of rotational correlation functions based on Brownian motion for complex disordered systems in a stochastic framework. We found out that the rotational relaxation function has a fractional form for complex disordered systems, which indicates that relaxation has nonexponential character and obeys the Kohlrausch-William-Watts law, following the Mittag-Leffler decay. PMID:15903722
Fokker-Planck description of the scattering of radio frequency waves at the plasma edge
Hizanidis, Kyriakos; Kominis, Yannis; Tsironis, Christos; Ram, Abhay K.
2010-02-15
In magnetic fusion devices, radio frequency (rf) waves in the electron cyclotron (EC) and lower hybrid (LH) range of frequencies are being commonly used to modify the plasma current profile. In ITER, EC waves are expected to stabilize the neoclassical tearing mode (NTM) by providing current in the island region [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)]. The appearance of NTMs severely limits the plasma pressure and leads to the degradation of plasma confinement. LH waves could be used in ITER to modify the current profile closer to the edge of the plasma. These rf waves propagate from the excitation structures to the core of the plasma through an edge region, which is characterized by turbulence--in particular, density fluctuations. These fluctuations, in the form of blobs, can modify the propagation properties of the waves by refraction. In this paper, the effect on rf due to randomly distributed blobs in the edge region is studied. The waves are represented as geometric optics rays and the refractive scattering from a distribution of blobs is formulated as a Fokker-Planck equation. The scattering can have two diffusive effects--one in real space and the other in wave vector space. The scattering can modify the trajectory of rays into the plasma and it can affect the wave vector spectrum. The refraction of EC waves, for example, could make them miss the intended target region where the NTMs occur. The broadening of the wave vector spectrum could broaden the wave generated current profile. The Fokker-Planck formalism for diffusion in real space and wave vector space is used to study the effect of density blobs on EC and LH waves in an ITER type of plasma environment. For EC waves the refractive effects become important since the distance of propagation from the edge to the core in ITER is of the order of a meter. The diffusion in wave vector space is small. For LH waves the refractive effects are insignificant but the diffusion in wave vector space is
Feedback-induced bistability of an optically levitated nanoparticle: A Fokker-Planck treatment
NASA Astrophysics Data System (ADS)
Ge, Wenchao; Rodenburg, Brandon; Bhattacharya, M.
2016-08-01
Optically levitated nanoparticles have recently emerged as versatile platforms for investigating macroscopic quantum mechanics and enabling ultrasensitive metrology. In this paper we theoretically consider two damping regimes of an optically levitated nanoparticle cooled by cavityless parametric feedback. Our treatment is based on a generalized Fokker-Planck equation derived from the quantum master equation presented recently and shown to agree very well with experiment [B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, and M. Bhattacharya, Quantum model of cooling and force sensing with an optically trapped nanoparticle, Optica 3, 318 (2016), 10.1364/OPTICA.3.000318]. For low damping, we find that the resulting Wigner function yields the single-peaked oscillator position distribution and recovers the appropriate energy distribution derived earlier using a classical theory and verified experimentally [J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state, Nat. Nano. 9, 358 (2014), 10.1038/nnano.2014.40]. For high damping, in contrast, we predict a double-peaked position distribution, which we trace to an underlying bistability induced by feedback. Unlike in cavity-based optomechanics, stochastic processes play a major role in determining the bistable behavior. To support our conclusions, we present analytical expressions as well as numerical simulations using the truncated Wigner function approach. Our work opens up the prospect of developing bistability-based devices, characterization of phase-space dynamics, and investigation of the quantum-classical transition using levitated nanoparticles.
Nonlinear Ehrenfest's urn model.
Casas, G A; Nobre, F D; Curado, E M F
2015-04-01
Ehrenfest's urn model is modified by introducing nonlinear terms in the associated transition probabilities. It is shown that these modifications lead, in the continuous limit, to a Fokker-Planck equation characterized by two competing diffusion terms, namely, the usual linear one and a nonlinear diffusion term typical of anomalous diffusion. By considering a generalized H theorem, the associated entropy is calculated, resulting in a sum of Boltzmann-Gibbs and Tsallis entropic forms. It is shown that the stationary state of the associated Fokker-Planck equation satisfies precisely the same equation obtained by extremization of the entropy. Moreover, the effects of the nonlinear contributions on the entropy production phenomenon are also analyzed. PMID:25974470
NASA Technical Reports Server (NTRS)
Khazanov, George V.; Khabibrakhmanov, Ildar K.; Glocer, Alex
2012-01-01
We present the results of a finite difference implementation of the kinetic Fokker-Planck model with an exact form of the nonlinear collisional operator, The model is time dependent and three-dimensional; one spatial dimension and two in velocity space. The spatial dimension is aligned with the local magnetic field, and the velocity space is defined by the magnitude of the velocity and the cosine of pitch angle. An important new feature of model, the concept of integration along the particle trajectories, is discussed in detail. Integration along the trajectories combined with the operator time splitting technique results in a solution scheme which accurately accounts for both the fast convection of the particles along the magnetic field lines and relatively slow collisional process. We present several tests of the model's performance and also discuss simulation results of the evolution of the plasma distribution for realistic conditions in Earth's plasmasphere under different scenarios.
NASA Astrophysics Data System (ADS)
Mohammadi, Masoumeh; Borzì, Alfio
2016-07-01
The Hermite spectral approximation of a hyperbolic Fokker-Planck (FP) optimality system arising in the control of an unbounded piecewise-deterministic process (PDP) is discussed. To control the probability density function (PDF) corresponding to the PDP process, an optimal control based on an FP strategy is considered. The resulting optimality system consists of a hyperbolic system with opposite-time orientation and an integral optimality condition equation. A Hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains. It is proven that the proposed scheme satisfies the conservativity requirement of the PDFs. The spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments.
Quasilinear simulation of auroral kilometric radiation by a relativistic Fokker-Planck code
Matsuda, Y.
1991-01-01
An intense terrestrial radiation called the auroral kilometric radiation (AKR) is believed to be generated by cyclotron maser instability. We study a quasilinear evolution of this instability by means of a two-dimensional relativistic Fokker-Planck code which treats waves and distributions self-consistently, including radiation loss and electron source and sink. We compare the distributions and wave amplitude with spacecraft observations to elucidate physical processes involved. 3 refs., 1 fig.
Variance reduction for Fokker-Planck based particle Monte Carlo schemes
NASA Astrophysics Data System (ADS)
Gorji, M. Hossein; Andric, Nemanja; Jenny, Patrick
2015-08-01
Recently, Fokker-Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1-3]. In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker-Planck model is considered. First, deviational based schemes were derived and reviewed, and it is shown that these deviational methods are not appropriate for practical Fokker-Planck based rarefied gas flow simulations. This is due to the fact that the deviational schemes considered in this study lead either to instabilities in the case of two-weight methods or to large statistical errors if the direct sampling method is applied. Motivated by this conclusion, we developed a novel scheme based on correlated stochastic processes. The main idea here is to synthesize an additional stochastic process with a known solution, which is simultaneously solved together with the main one. By correlating the two processes, the statistical errors can dramatically be reduced; especially for low Mach numbers. To assess the methods, homogeneous relaxation, planar Couette and lid-driven cavity flows were considered. For these test cases, it could be demonstrated that variance reduction based on parallel processes is very robust and effective.
NASA Astrophysics Data System (ADS)
Lin, XiaoHui; Zhang, ChiBin; Gu, Jun; Jiang, ShuYun; Yang, JueKuan
2014-11-01
A non-continuous electroosmotic flow model (PFP model) is built based on Poisson equation, Fokker-Planck equation and Navier-Stokse equation, and used to predict the DNA molecule translocation through nanopore. PFP model discards the continuum assumption of ion translocation and considers ions as discrete particles. In addition, this model includes the contributions of Coulomb electrostatic potential between ions, Brownian motion of ions and viscous friction to ion transportation. No ionic diffusion coefficient and other phenomenological parameters are needed in the PFP model. It is worth noting that the PFP model can describe non-equilibrium electroosmotic transportation of ions in a channel of a size comparable with the mean free path of ion. A modified clustering method is proposed for the numerical solution of PFP model, and ion current translocation through nanopore with a radius of 1 nm is simulated using the modified clustering method. The external electric field, wall charge density of nanopore, surface charge density of DNA, as well as ion average number density, influence the electroosmotic velocity profile of electrolyte solution, the velocity of DNA translocation through nanopore and ion current blockade. Results show that the ion average number density of electrolyte and surface charge density of nanopore have a significant effect on the translocation velocity of DNA and the ion current blockade. The translocation velocity of DNA is proportional to the surface charge density of nanopore, and is inversely proportional to ion average number density of electrolyte solution. Thus, the translocation velocity of DNAs can be controlled to improve the accuracy of sequencing by adjusting the external electric field, ion average number density of electrolyte and surface charge density of nanopore. Ion current decreases when the ion average number density is larger than the critical value and increases when the ion average number density is lower than the
NASA Astrophysics Data System (ADS)
Dominguez, Efrain; Rosmann, Thomas; Chavarro, John
2014-05-01
In order to extract the mathematical operators that rule complex system behavior, a numeric scheme of the multidimensional Fokker-Planck-Kolmogorov equation is proposed allowing, through conjugate gradient optimization, the identification of deterministic kernels for an observed complex system. This scheme is analyzed using a hydrological basin as example but can be used in many fields. It is assumed that there are observed input-output signals of the system and no especial assumptions about the system kernel are required. This approach can be used at different time resolutions and it is expected to be powerful enough to characterize hydrological variability at different time scales, even under no-stationary conditions. This inverse modeling scheme has three different identification methods, the first one is related to Langevin equations system types, thus random components are described, additively, as noises while in the second method they are represented by the noises intensities instead of noise processes itself. As a result of this inverse modeling approach, hydrological processes can be described as a combination of deterministic kernels and random processes and the system phase space dimensionality can be objectively established. In this work, proposed approach was used to study hydrological variability, trends and extremes at different time resolution.
Modeling the dynamics of charged-particle beams using a Fokker-Planck approach
Bohn, C.L.; Delayen, J.R.
1994-10-01
At the previous Linac Conference, we introduced a semianalytic Fokker-Planck formalism for calculating the evolution of intense, nonrelativistic, mismatched beams propagating through focusing channels. We have since elaborated on its physical basis and greatly expanded its applicability. In this paper, we implement the model to study the dynamics of a circularly symmetric beam propagating through a linear focusing channel. An example is discussed for an ion beam and accelerator parameters which are representative of high-current spallation neutron sources in which space-charge forces are important. The example illustrates the dynamics of emittance growth and halo formation in the beam.
NASA Astrophysics Data System (ADS)
Taguchi, Masayoshi
1982-05-01
The neoclassical transport coefficients from the plateau regime to the Pfirsch-Schlüter regime are calculated by using the linearized full Fokker-Planck collision operator. The inverse aspect ratio is assumed to be small and the mass ratio of the electron to the ion is much less than unity. The lowest-order terms of these ratios are retained in this calculation. The results obtained give correction to those of Rawls et al. who employed a model collision operator. In the Pfirsch-Schlüter regime, the results of this paper agree fairly well with those given by Hazeltine and Hinton.
Fokker-Planck Modelling of Delayed Loss of Charged Fusion Products in TFTR.
Edenstrasser, J.W.; Goloborod'ko, V.Ya.; Reznik, S.N.; Yavorskij, V.A.; Zweben, S.
1998-08-01
The results of a Fokker-Planck simulation of the ripple-induced loss of charged fusion products in the Tokamak Fusion Test Reactor (TFTR) are presented. It is shown that the main features of the measured "delayed loss" of partially thermalized fusion products, such as the differences between deuterium-deuterium and deuterium-tritium discharges, the plasma current and major radius dependencies, etc., are in satisfactory agreement with the classical collisional ripple transport mechanism. The inclusion of the inward shift of the vacuum flux surfaces turns out to be necessary for an adequate and consistent explanation of the origin of the partially thermalized fusion product loss to the bottom of TFTR.
Coupled Ray-tracing and Fokker-Planck EBW Modeling for Spherical Tokamaks
NASA Astrophysics Data System (ADS)
Urban, J.; Decker, J.; Peysson, Y.; Preinhaelter, J.; Taylor, G.; Vahala, L.; Vahala, G.
2009-11-01
The AMR (Antenna—Mode-conversion—Ray-tracing) code [1, 2] has been recently coupled with the LUKE [3] Fokker-Planck code. This modeling suite is capable of complex simulations of electron Bernstein wave (EBW) emission, heating and current drive. We employ these codes to study EBW heating and current drive performance under spherical tokamak (ST) configurations—typical NSTX discharges are employed. EBW parameters, such as frequency, antenna position and direction, are varied and optimized for particular configurations and objectives. In this way, we show the versatility of EBWs.
NASA Technical Reports Server (NTRS)
Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1975-01-01
The Fokker-Planck coefficient for pitch-angle scattering, appropriate for cosmic rays in homogeneous stationary magnetic turbulence is computed without making any specific assumptions concerning the statistical symmetries of the random field. The Fokker-Planck coefficient obtained can be used to compute the parallel diffusion coefficient for high-energy cosmic rays propagating in the presence of strong turbulence, or for low-energy cosmic rays in the presence of weak turbulence. Because of the generality of magnetic turbulence allowed for in the analysis, special interplanetary magnetic field features, such as discontinuities or particular wave modes, can be included rigorously.
Sloan, D.P.
1983-05-01
Morel (1981) has developed multigroup Legendre cross sections suitable for input to standard discrete ordinates transport codes for performing charged-particle Fokker-Planck calculations in one-dimensional slab and spherical geometries. Since the Monte Carlo neutron transport code, MORSE, uses the same multigroup cross section data that discrete ordinates codes use, it was natural to consider whether Fokker-Planck calculations could be performed with MORSE. In order to extend the unique three-dimensional forward or adjoint capability of MORSE to Fokker-Planck calculations, the MORSE code was modified to correctly treat the delta-function scattering of the energy operator, and a new set of physically acceptable cross sections was derived to model the angular operator. Morel (1979) has also developed multigroup Legendre cross sections suitable for input to standard discrete ordinates codes for performing electron Boltzmann calculations. These electron cross sections may be treated in MORSE with the same methods developed to treat the Fokker-Planck cross sections. The large magnitude of the elastic scattering cross section, however, severely increases the computation or run time. It is well-known that approximate elastic cross sections are easily obtained by applying the extended transport (or delta function) correction to the Legendre coefficients of the exact cross section. An exact method for performing the extended transport cross section correction produces cross sections which are physically acceptable. Sample calculations using electron cross sections have demonstrated this new technique to be very effective in decreasing the large magnitude of the cross sections.
Conceptual foundation of the Fokker-Planck approach to space-charge effects
Bohn, C.L.
1996-07-01
An rms-mismatched beam can evolve rapidly to a configuration of quasiequilibrium under the influence of space-charge forces. As sit evolves, its emittance grows and a diffuse halo forms. The beam`s distribution function accounts for all the complicated dynamics. Unfortunately, the distribution function is difficult to calculate inasmuch as the physics lies at the interface between classical mechanics and thermodynamics. This paper presents the foundation for a statistical theory of the dynamics of nonequilibrium space-charge-dominated beams. Within certain approximations, the theory takes on a Fokker-Planck form. Key questions arise concerning the nature of the dynamical friction and diffusion in the beam`s phase space and of the quasiequilibrium configuration that ensues.
A Fokker-Planck code for laser plasma interaction in femtosecond-laser shock peening
NASA Astrophysics Data System (ADS)
Ren, Zhencheng; Wang, Guo-Xiang; Ye, Chang; Dong, Yalin
2016-03-01
A Fokker-Planck code is developed to simulate the laser-plasma interaction in the femtosecond-laser shock peening and forming processes. A numerical scheme dealing with high-energy concentration and its resulting steep gradient are presented, and the source code is provided as supplementary material for further usage. The breakdown of the classical heat transport theory is observed when the laser intensity increases. The difference in heat flow between the classical theory and simulation is presented. It is found that the classical heat transport theory overestimates heat flow by orders of magnitude during femtosecond-laser shock peening or forming. As a result, the electron pressure can be underestimated using the classical hydrodynamic code.
A method for the analysis of nonlinearities in aircraft dynamic response to atmospheric turbulence
NASA Technical Reports Server (NTRS)
Sidwell, K.
1976-01-01
An analytical method is developed which combines the equivalent linearization technique for the analysis of the response of nonlinear dynamic systems with the amplitude modulated random process (Press model) for atmospheric turbulence. The method is initially applied to a bilinear spring system. The analysis of the response shows good agreement with exact results obtained by the Fokker-Planck equation. The method is then applied to an example of control-surface displacement limiting in an aircraft with a pitch-hold autopilot.
Non-Markov stochastic processes satisfying equations usually associated with a Markov process
NASA Astrophysics Data System (ADS)
McCauley, J. L.
2012-04-01
There are non-Markov Ito processes that satisfy the Fokker-Planck, backward time Kolmogorov, and Chapman-Kolmogorov equations. These processes are non-Markov in that they may remember an initial condition formed at the start of the ensemble. Some may even admit 1-point densities that satisfy a nonlinear 1-point diffusion equation. However, these processes are linear, the Fokker-Planck equation for the conditional density (the 2-point density) is linear. The memory may be in the drift coefficient (representing a flow), in the diffusion coefficient, or in both. We illustrate the phenomena via exactly solvable examples. In the last section we show how such memory may appear in cooperative phenomena.
Nonstationary probability densities of a class of nonlinear system excited by external colored noise
NASA Astrophysics Data System (ADS)
Qi, LuYuan; Xu, Wei; Gu, XuDong
2012-03-01
This paper deals with the approximate nonstationary probability density of a class of nonlinear vibrating system excited by colored noise. First, the stochastic averaging method is adopted to obtain the averaged Itô equation for the amplitude of the system. The corresponding Fokker-Planck-Kolmogorov equation governing the evolutionary probability density function is deduced. Then, the approximate solution of the Fokker-Planck-Kolmogorov equation is derived by applying the Galerkin method. The solution is expressed as a sum of a series of expansion in terms of a set of proper basis functions with time-depended coefficients. Finally, an example is given to illustrate the proposed procedure. The validity of the proposed method is confirmed by Monte Carlo Simulation.
NASA Technical Reports Server (NTRS)
Fisk, L. A.; Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1973-01-01
For the case of homogeneous, isotropic magnetic field fluctuations, it is shown that most theories which are based on the quasi-linear and adiabatic approximation yield the same integral for the Fokker-Planck coefficient for the pitch angle scattering of cosmic rays. For example, despite apparent differences, the theories due to Jokipii and to Klimas and Sandri yield the same integral. It is also shown, however, that this integral in most cases has been evaluated incorrectly in the past. For large pitch angles these errors become significant, and for pitch angles of 90 deg the actual Fokker-Planck coefficient contains a delta function. The implications for these corrections relating cosmic ray diffusion coefficients to observed properties of the interplanetary magnetic field are discussed.
NASA Technical Reports Server (NTRS)
Goldstein, M. L.; Klimas, A. J.; Sandri, G.
1974-01-01
The Fokker-Planck coefficient for pitch angle scattering, appropriate for cosmic rays in homogeneous, stationary, magnetic turbulence, is computed from first principles. No assumptions are made concerning any special statistical symmetries the random field may have. This result can be used to compute the parallel diffusion coefficient for high energy cosmic rays moving in strong turbulence, or low energy cosmic rays moving in weak turbulence. Becuase of the generality of the magnetic turbulence which is allowed in this calculation, special interplanetary magnetic field features such as discontinuities, or particular wave modes, can be included rigorously. The reduction of this results to previously available expressions for the pitch angle scattering coefficient in random field models with special symmetries is discussed. The general existance of a Dirac delta function in the pitch angle scattering coefficient is demonstrated. It is proved that this delta function is the Fokker-Planck prediction for pitch angle scattering due to mirroring in the magnetic field.
Fokker-Planck simulation of runaway electron generation in disruptions with the hot-tail effect
NASA Astrophysics Data System (ADS)
Nuga, H.; Yagi, M.; Fukuyama, A.
2016-06-01
To study runaway electron generation in disruptions, we have extended the three-dimensional (two-dimensional in momentum space; one-dimensional in the radial direction) Fokker-Planck code, which describes the evolution of the relativistic momentum distribution function of electrons and the induced toroidal electric field in a self-consistent manner. A particular focus is placed on the hot-tail effect in two-dimensional momentum space. The effect appears if the drop of the background plasma temperature is sufficiently rapid compared with the electron-electron slowing down time for a few times of the pre-quench thermal velocity. It contributes to not only the enhancement of the primary runaway electron generation but also the broadening of the runaway electron distribution in the pitch angle direction. If the thermal energy loss during the major disruption is assumed to be isotropic, there are hot-tail electrons that have sufficiently large perpendicular momentum, and the runaway electron distribution becomes broader in the pitch angle direction. In addition, the pitch angle scattering also yields the broadening. Since the electric field is reduced due to the burst of runaway electron generation, the time required for accelerating electrons to the runaway region becomes longer. The longer acceleration period makes the pitch-angle scattering more effective.
Loading, absorption, and Fokker-Planck calculations for upcoming ICRF experiments on ATF
Shepard, T.D.; Carter, M.D.; Goulding, R.H.; Kwon, M.
1989-01-01
ICRF experiments on ATF at the 100-kW level are planned for the current 1989 operating period. These plans include the 2..omega../sub cH/ regime at f/sub RF/ = 28.88 MHz, D(H) at 14.44 MHz, and /sup 4/He(/sup 3/He) and D(/sup 3/He) at 9.63 MHz. ECH target plasmas have n/sub eO/ /approx lt/ 0.15 /times/ 10/sup 20/ m/sup /minus/3/ and B = 0.95 T. The density and temperature profiles obtained are broader than those from 1988, owing to recent field error corrections. The values used for target-plasma parameters in the calculations were taken from initial 1989 ATF data. Loading and absorption calculations have been performed using the 3D RF heating code ORION with a helically symmetric equilibrium, and Fokker-Planck calculations were performed using the steady-state code RFTRANS with two velocity dimensions and one spatial dimension. 6 refs., 3 figs.
A finite volume Fokker-Planck collision operator in constants-of-motion coordinates
NASA Astrophysics Data System (ADS)
Xiong, Z.; Xu, X. Q.; Cohen, B. I.; Cohen, R.; Dorr, M. R.; Hittinger, J. A.; Kerbel, G.; Nevins, W. M.; Rognlien, T.
2006-04-01
TEMPEST is a 5D gyrokinetic continuum code for edge plasmas. Constants of motion, namely, the total energy E and the magnetic moment μ, are chosen as coordinate s because of their advantage in minimizing numerical diffusion in advection operato rs. Most existing collision operators are written in other coordinates; using them by interpolating is shown to be less satisfactory in maintaining overall numerical accuracy and conservation. Here we develop a Fokker-Planck collision operator directly in (E,μ) space usin g a finite volume approach. The (E, μ) grid is Cartesian, and the turning point boundary represents a straight line cutting through the grid that separates the ph ysical and non-physical zones. The resulting cut-cells are treated by a cell-mergin g technique to ensure a complete particle conservation. A two dimensional fourth or der reconstruction scheme is devised to achieve good numerical accuracy with modest number of grid points. The new collision operator will be benchmarked by numerical examples.
Wang, Chi-Jen
2013-01-01
In this thesis, we analyze both the spatiotemporal behavior of: (A) non-linear “reaction” models utilizing (discrete) reaction-diffusion equations; and (B) spatial transport problems on surfaces and in nanopores utilizing the relevant (continuum) diffusion or Fokker-Planck equations. Thus, there are some common themes in these studies, as they all involve partial differential equations or their discrete analogues which incorporate a description of diffusion-type processes. However, there are also some qualitative differences, as shall be discussed below.
Bizarro, J.P.; Peysson, Y.; Bonoli, P.T.; Carrasco, J.; de Wit, T.D.; Fuchs, V.; Hoang, G.T.; Litaudon, X.; Moreau, D.; Pocheau, C.; Shkarofsky, I.P. )
1993-09-01
A detailed investigation is presented on the ability of combined ray-tracing and Fokker--Planck calculations to predict the hard x-ray (HXR) emission during lower-hybrid (LH) current drive in tokamaks when toroidally induced ray stochasticity is important. A large number of rays is used and the electron distribution function is obtained by self-consistently iterating the appropriate power deposition and Fokker--Planck calculations. It is shown that effects due to radial diffusion of suprathermal electrons and to radiation scattering by the inner wall can be significant. The experimentally observed features of the HXR emission are fairly well predicted, thus confirming that combined ray-tracing and Fokker--Planck codes are capable of correctly modeling the physics of LH current drive in tokamaks.
Heikkinen, J.A. ); Paettikangas, T.J.H. )
1994-09-01
The evolution of a one-dimensional velocity distribution is studied in the presence of a monochromatic large-amplitude periodic force which is turned on adiabatically. The periodic Vlasov-Poisson equations are solved in the presence of a linearized Fokker-Planck collision term. For a constant driving force, the system is found to approach, after transient oscillations, a steady state which is maintained by one wave at the driving frequency. This is in contrast to the result in the absence of collisions where the steady state tends to be supported by several waves. An analytical solution for the steady-state distribution function in the presence of a driven large-amplitude wave is obtained by a Hamiltonian approach. The distribution function is expanded in powers of a small parameter [Gamma] proportional to the collision strength. From the expansion, the zeroth order term is shown to give the space-averaged distribution function correct to first order in [Gamma]. Comparison with the results of the simulations and of the harmonics expansion method shows that the solution estimates the distribution with good accuracy. The plateau in the wave trapping regime is analyzed, and the current driven by the large-amplitude traveling wave is determined.
NASA Astrophysics Data System (ADS)
Joglekar, Archis; Thomas, Alexander; Read, Martin; Kingham, Robert
2014-10-01
Here, we present 2D numerical modeling of near critical density plasma using a fully implicit Vlasov-Fokker-Planck (VFP) code, IMPACTA, with the addition of a ray tracing package. In certain situations, such as those at the critical surface at the walls of a hohlraum, magnetic fields are generated through the crossed temperature and electron density gradients. Modeling shows 0.3 MG fields and the strong heating also results in magnetization of the plasma up to ωτ ~ 5 . In the case without magnetic field generation, the heat flows from the laser heating region are isotropic. Including magnetic fields causes the heat flow to form jets along the wall due to the Righi-Leduc effect. The heating of the wall region causes steeper temperature gradients. This serves as a positive feedback mechanism for the field generation rate resulting in nearly twice the amount of field generated in comparison to the case without magnetic fields over 1 ns. The heat conduction, field generation, and the calculation of other transport quantities, is performed ab-initio due to the nature of the VFP equation set. In order to determine the importance of the kinetic effects from IMPACTA, we perform direct comparison with a classical (Braginskii) transport code with hydrodynamic motion (CTC+). The authors would like to acknowledge DOE Grant #DESC0010621 and Advanced Research Computing, UM-AA.
Tzoufras, M.; Tableman, A.; Tsung, F. S.; Mori, W. B.; Bell, A. R.
2013-05-15
To study the kinetic physics of High-Energy-Density Laboratory Plasmas, we have developed the parallel relativistic 2D3P Vlasov-Fokker-Planck code Oshun. The numerical scheme uses a Cartesian mesh in configuration-space and incorporates a spherical harmonic expansion of the electron distribution function in momentum-space. The expansion is truncated such that the necessary angular resolution of the distribution function is retained for a given problem. Finite collisionality causes rapid decay of the high-order harmonics, thereby providing a natural truncation mechanism for the expansion. The code has both fully explicit and implicit field-solvers and employs a linearized Fokker-Planck collision operator. Oshun has been benchmarked against well-known problems, in the highly kinetic limit to model collisionless relativistic instabilities, and in the hydrodynamic limit to recover transport coefficients. The performance of the code, its applicability, and its limitations are discussed in the context of simple problems with relevance to inertial fusion energy.
NASA Astrophysics Data System (ADS)
Colombant, Denis; Manheimer, Wallace
2008-11-01
The Krook model described in the previous talk has been incorporated into a fluid simulation. These fluid simulations are then compared with Fokker Planck simulations and also with a recent NRL Nike experiment. We also examine several other models for electron energy transport that have been used in laser fusion research. As regards comparison with Fokker Planck simulation, the Krook model gives better agreement than the other models, especially in the time asymptotic limit. As regards the NRL experiment, all models except one give reasonable agreement.
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation
NASA Astrophysics Data System (ADS)
Mischler, S.; Mouhot, C.
2016-08-01
The aim of the present paper is twofold: 1. We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the "mild perturbation" part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series.
The Dynamics of M15: Observations of the Velocity Dispersion Profile and Fokker-Planck Models
NASA Astrophysics Data System (ADS)
Dull, J. D.; Cohn, H. N.; Lugger, P. M.; Murphy, B. W.; Seitzer, P. O.; Callanan, P. J.; Rutten, R. G. M.; Charles, P. A.
1997-05-01
We report a new measurement of the velocity dispersion profile within 1' (3 pc) of the center of the globular cluster M15 (NGC 7078), using long-slit spectra from the 4.2 m William Herschel Telescope at La Palma Observatory. We obtained spatially resolved spectra for a total of 23 slit positions during two observing runs. During each run, a set of parallel slit positions was used to map out the central region of the cluster; the position angle used during the second run was orthogonal to that used for the first. The spectra are centered in wavelength near the Ca II infrared triplet at 8650 Å, with a spectral range of about 450 Å. We determined radial velocities by cross-correlation techniques for 131 cluster members. A total of 32 stars were observed more than once. Internal and external comparisons indicate a velocity accuracy of about 4 km s-1. The velocity dispersion profile rises from about σ = 7.2 +/- 1.4 km s-1 near 1' from the center of the cluster to σ = 13.9 +/- 1.8 km s-1 at 20". Inside of 20", the dispersion remains approximately constant at about 10.2 +/- 1.4 km s-1 with no evidence for a sharp rise near the center. This last result stands in contrast with that of Peterson, Seitzer, & Cudworth who found a central velocity dispersion of 25 +/- 7 km s-1, based on a line-broadening measurement. Our velocity dispersion profile is in good agreement with those determined in the recent studies of Gebhardt et al. and Dubath & Meylan. We have developed a new set of Fokker-Planck models and have fitted these to the surface brightness and velocity dispersion profiles of M15. We also use the two measured millisecond pulsar accelerations as constraints. The best-fitting model has a mass function slope of x = 0.9 (where 1.35 is the slope of the Salpeter mass function) and a total mass of 4.9 × 105 M⊙. This model contains approximately 104 neutron stars (3% of the total mass), the majority of which lie within 6" (0.2 pc) of the cluster center. Since the
Fokker-Planck/Ray Tracing for Electron Bernstein and Fast Wave Modeling in Support of NSTX
Harvey, R. W.
2009-11-12
This DOE grant supported fusion energy research, a potential long-term solution to the world's energy needs. Magnetic fusion, exemplified by confinement of very hot ionized gases, i.e., plasmas, in donut-shaped tokamak vessels is a leading approach for this energy source. Thus far, a mixture of hydrogen isotopes has produced 10's of megawatts of fusion power for seconds in a tokamak reactor at Princeton Plasma Physics Laboratory in New Jersey. The research grant under consideration, ER54684, uses computer models to aid in understanding and projecting efficacy of heating and current drive sources in the National Spherical Torus Experiment, a tokamak variant, at PPPL. The NSTX experiment explores the physics of very tight aspect ratio, almost spherical tokamaks, aiming at producing steady-state fusion plasmas. The current drive is an integral part of the steady-state concept, maintaining the magnetic geometry in the steady-state tokamak. CompX further developed and applied models for radiofrequency (rf) heating and current drive for applications to NSTX. These models build on a 30 year development of rf ray tracing (the all-frequencies GENRAY code) and higher dimensional Fokker-Planck rf-collisional modeling (the 3D collisional-quasilinear CQL3D code) at CompX. Two mainline current-drive rf modes are proposed for injection into NSTX: (1) electron Bernstein wave (EBW), and (2) high harmonic fast wave (HHFW) modes. Both these current drive systems provide a means for the rf to access the especially high density plasma--termed high beta plasma--compared to the strength of the required magnetic fields. The CompX studies entailed detailed modeling of the EBW to calculate the efficiency of the current drive system, and to determine its range of flexibility for driving current at spatial locations in the plasma cross-section. The ray tracing showed penetration into NSTX bulk plasma, relatively efficient current drive, but a limited ability to produce current over the whole
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
NASA Technical Reports Server (NTRS)
Hizanidis, Kyriakos; Vlahos, L.; Polymilis, C.
1989-01-01
The relativistic motion of an ensemble of electrons in an intense monochromatic electromagnetic wave propagating obliquely in a uniform external magnetic field is studied. The problem is formulated from the viewpoint of Hamiltonian theory and the Fokker-Planck-Kolmogorov approach analyzed by Hizanidis (1989), leading to a one-dimensional diffusive acceleration along paths of constant zeroth-order generalized Hamiltonian. For values of the wave amplitude and the propagating angle inside the analytically predicted stochastic region, the numerical results suggest that the diffusion probes proceeds in stages. In the first stage, the electrons are accelerated to relatively high energies by sampling the first few overlapping resonances one by one. During that stage, the ensemble-average square deviation of the variable involved scales quadratically with time. During the second stage, they scale linearly with time. For much longer times, deviation from linear scaling slowly sets in.
Neoclassical Transport Including Collisional Nonlinearity
Candy, J.; Belli, E. A.
2011-06-10
In the standard {delta}f theory of neoclassical transport, the zeroth-order (Maxwellian) solution is obtained analytically via the solution of a nonlinear equation. The first-order correction {delta}f is subsequently computed as the solution of a linear, inhomogeneous equation that includes the linearized Fokker-Planck collision operator. This equation admits analytic solutions only in extreme asymptotic limits (banana, plateau, Pfirsch-Schlueter), and so must be solved numerically for realistic plasma parameters. Recently, numerical codes have appeared which attempt to compute the total distribution f more accurately than in the standard ordering by retaining some nonlinear terms related to finite-orbit width, while simultaneously reusing some form of the linearized collision operator. In this work we show that higher-order corrections to the distribution function may be unphysical if collisional nonlinearities are ignored.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Physics-based computational complexity of nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Frederick E.; Huang, Jim
2004-08-01
Our theory is based on the mapping between two Fokker-Planck equations and two Schroedinger equations (see [1] & [2]), which is well known in physics, but which has not been exploited in filtering theory. This theory expands Brockett's Lie algebra homomorphism conjecture for characterizing finite dimensional filters. In particular, the Schroedinger equation generates a group, whereas the Zakai equation (as well as the Fokker-Planck equation) does not, owing to the lack of a smooth inverse. Simple non-pathological low-dimensional linear-Gaussian timeinvariant counterexamples show that Brockett's conjecture does not reliably predict when a nonlinear filtering problem will have an exact finite dimensional solution. That is, there are manifestly finite dimensional filters for estimation problems with infinite dimensional Lie algebras. There are three reasons that the Lie algebraic approach as originally formulated by Brockett is incomplete: (1) the Zakai equation does not generate a group; (2) Lie algebras are coordinate free, whereas separation of variables in PDEs is not coordinate free, and (3) Brockett's theory aims to characterize finite dimensional filters for any initial condition of the Zakai equation, whereas SOV for PDEs generally depends on the initial condition. We will attempt to make this paper accessible to normal engineers who do not have Lie algebras for breakfast.
Comparison of Finite Element Non-Linear Beam Random Response with Experimental Results
NASA Astrophysics Data System (ADS)
Chen, R. R.; Mei, C.; Wolfe, HF
1996-09-01
A finite element formulation combined with the equivalent linearization technique and normal mode method is developed for the non-linear random response of beams subjected to acoustic and thermal loads applied simultaneously. To validate the present formulation and solution procedure, results are compared with the classical continuum solution and the Fokker-Planck-Kolmogorov equation solution. Comparison is also made with experimental data for a pre-stretched clamped beam. Random responses of thermally buckled simply supported beam, clamped beam and simply supported-clamped beam are presented. The comparison of the present simultaneously loaded response with the existing sequentially loaded results shows a significant difference between them.
NASA Astrophysics Data System (ADS)
Chen, Ruixi; Mei, Chuh
1993-04-01
A finite element formulation combined with the equivalent linearization technique and the normal mode method is developed for the study of nonlinear random response of beams subjected to simultaneously applied acoustic and thermal loads. Examples include thermally buckled random response of simply supported beam, clamped-clamped beam and simply supported-clamped beam. To compare and validate the present formulation, results are compared with the solutions from existing sequential load method, and significant difference has been found. Results by classical continuum solution and the solution of Fokker-Planck-Kolmogorov equation are also derived and obtained for comparison.
Keanini, R.G.
2011-04-15
Research Highlights: > Systematic approach for physically probing nonlinear and random evolution problems. > Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. > Organization of near-molecular scale vorticity mediated by hydrodynamic modes. > Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion
NASA Astrophysics Data System (ADS)
Gnesin, Silvano; Goodman, Timothy; Coda, Stefano; Decker, Joan; Peysson, Yves
2010-11-01
The TCV tokamak is equipped with nine electron cyclotron (EC) wave gyrotron/launcher systems: six 0.5 MW in the 2nd harmonic X-mode (X2) and three 0.5 MW in the 3rd harmonic X-mode (X3). TCV experiments have been expressly devised to study the X2/X3 interplay, especially through the dynamics and transport properties of the suprathermal electron population generated primarily by X2 and its influence on the X3 wave absorption. Fokker Planck modeling of X2/X3 TCV experiments with the quasilinear fully relativistic LUKE code, coupled with the C3PO ray-tracing module and the R5X2 bremsstrahlung module, is presented here. Two series of experiments are discussed: 1) X2/X3 synergy when both X2 (82.7 GHz) and X3 (118 GHz) waves are injected into the plasma and 2) X2/X3 synergetic absorption at the same frequency (82.7 GHz). The role of suprathermal electron transport has been investigated by comparing the bremsstrahlung emission measured by a hard X-ray camera with the simulated signal.
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method
NASA Astrophysics Data System (ADS)
Er, GuoKang; Iu, VaiPan
2011-09-01
The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.
Mesh-free adjoint methods for nonlinear filters
NASA Astrophysics Data System (ADS)
Daum, Fred
2005-09-01
We apply a new industrial strength numerical approximation, called the "mesh-free adjoint method", to solve the nonlinear filtering problem. This algorithm exploits the smoothness of the problem, unlike particle filters, and hence we expect that mesh-free adjoints are superior to particle filters for many practical applications. The nonlinear filter problem is equivalent to solving the Fokker-Planck equation in real time. The key idea is to use a good adaptive non-uniform quantization of state space to approximate the solution of the Fokker-Planck equation. In particular, the adjoint method computes the location of the nodes in state space to minimize errors in the final answer. This use of an adjoint is analogous to optimal control algorithms, but it is more interesting. The adjoint method is also analogous to importance sampling in particle filters, but it is better for four reasons: (1) it exploits the smoothness of the problem; (2) it explicitly minimizes the errors in the relevant functional; (3) it explicitly models the dynamics in state space; and (4) it can be used to compute a corrected value for the desired functional using the residuals. We will attempt to make this paper accessible to normal engineers who do not have PDEs for breakfast.
NASA Astrophysics Data System (ADS)
Green, P. L.; Worden, K.; Atallah, K.; Sims, N. D.
2012-09-01
This work is concerned with the performance of a single degree of freedom electromagnetic energy harvester when subjected to a broadband white noise base acceleration. First, using the Fokker-Planck-Kolmogorov equation, it is shown that Duffing-type nonlinearities can be used to reduce the size of energy harvesting devices without affecting their power output. This is then verified using the technique of Equivalent Linearisation. Second, it is shown analytically that the optimum load resistance of the device is different to that which is dictated by the principle of impedance matching. This result is then verified experimentally.
Optimal feedback control of strongly non-linear systems excited by bounded noise
NASA Astrophysics Data System (ADS)
Zhu, W. Q.; Huang, Z. L.; Ko, J. M.; Ni, Y. Q.
2004-07-01
A strategy for non-linear stochastic optimal control of strongly non-linear systems subject to external and/or parametric excitations of bounded noise is proposed. A stochastic averaging procedure for strongly non-linear systems under external and/or parametric excitations of bounded noise is first developed. Then, the dynamical programming equation for non-linear stochastic optimal control of the system is derived from the averaged Itô equations by using the stochastic dynamical programming principle and solved to yield the optimal control law. The Fokker-Planck-Kolmogorov equation associated with the fully completed averaged Itô equations is solved to give the response of optimally controlled system. The application and effectiveness of the proposed control strategy are illustrated with the control of cable vibration in cable-stayed bridges and the feedback stabilization of the cable under parametric excitation of bounded noise.
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
NASA Astrophysics Data System (ADS)
Kazakevičius, R.; Ruseckas, J.
2015-11-01
Subdiffusive behavior of one-dimensional stochastic systems can be described by time-subordinated Langevin equations. The corresponding probability density satisfies the time-fractional Fokker-Planck equations. In the homogeneous systems the power spectral density of the signals generated by such Langevin equations has power-law dependency on the frequency with the exponent smaller than 1. In this paper we consider nonhomogeneous systems and show that in such systems the power spectral density can have power-law behavior with the exponent equal to or larger than 1 in a wide range of intermediate frequencies.
Electron dynamics with radiation and nonlinear wigglers
Jowett, J.M.
1986-06-01
The physics of electron motion in storage rings is described by supplementing the Hamiltonian equations of motion with fluctuating radiation reaction forces to describe the effects of synchrotron radiation. This leads to a description of radiation damping and quantum diffusion in single-particle phase-space by means of Fokker-Planck equations. For practical purposes, most storage rings remain in the regime of linear damping and diffusion; this is discussed in some detail with examples, concentrating on longitudinal phase space. However special devices such as nonlinear wigglers may permit the new generation of very large rings to go beyond this into regimes of nonlinear damping. It is shown how a special combined-function wiggler can be used to modify the energy distribution and current profile of electron bunches.
NASA Astrophysics Data System (ADS)
Barrett, John W.; Süli, Endre
2016-07-01
We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Ω in Rd, d = 2, for the density ρ, the velocity u ˜ and the pressure p of the fluid, with an equation of state of the form p (ρ) =cpργ, where cp is a positive constant and γ > 1. The right-hand side of the Navier-Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Süli (2016) [9], which established the existence of global-in-time weak solutions to the system for d ∈ { 2 , 3 } and γ >3/2, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d = 2 and γ > 1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density ρ0 ∈L∞ (Ω) for the continuity equation; a square-integrable initial velocity datum u˜0 for the Navier-Stokes momentum equation; and a nonnegative initial probability density function ψ0
Isostable reduction with applications to time-dependent partial differential equations.
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system. PMID:27575127
Isostable reduction with applications to time-dependent partial differential equations
NASA Astrophysics Data System (ADS)
Wilson, Dan; Moehlis, Jeff
2016-07-01
Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.
Nonlinear filtering for spacecraft attitude estimation
NASA Technical Reports Server (NTRS)
Vathsal, S.
1986-01-01
Nonlinear filtering techniques are applied to spacecraft attitude estimation using quaternion parameterization for the attitude kinematics. By replacing the angular velocity vector by the gyro output vector, a state dependent noise vector is introduced in the seven-dimensional system equations. The resulting conditional probability density function from the Ito differential rule is governed by the Fokker Planck partial differential equation which is approximated by the second order mean and covariance differential equations. In order to minimize computer loading, the covariance propagation is carried out in six-dimensional state space using a matrix transformation. The star tracker data is used to update the covariance matrix in the seven-dimensional space. The algorithm is simulated for an earth pointing spacecraft mission, using Monte Carlo samples of gyro and star measurements. The performance of the second order filter is compared with the extended Kalman Filter through several simulation runs and drift rates have been identified.
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
Coupled Particle Transport and Pattern Formation in a Nonlinear Leaky-Box Model
NASA Technical Reports Server (NTRS)
Barghouty, A. F.; El-Nemr, K. W.; Baird, J. K.
2009-01-01
Effects of particle-particle coupling on particle characteristics in nonlinear leaky-box type descriptions of the acceleration and transport of energetic particles in space plasmas are examined in the framework of a simple two-particle model based on the Fokker-Planck equation in momentum space. In this model, the two particles are assumed coupled via a common nonlinear source term. In analogy with a prototypical mathematical system of diffusion-driven instability, this work demonstrates that steady-state patterns with strong dependence on the magnetic turbulence but a rather weak one on the coupled particles attributes can emerge in solutions of a nonlinearly coupled leaky-box model. The insight gained from this simple model may be of wider use and significance to nonlinearly coupled leaky-box type descriptions in general.
The probabilistic solution of stochastic oscillators with even nonlinearity under poisson excitation
NASA Astrophysics Data System (ADS)
Guo, Siu-Siu; Er, Guo-Kang
2012-06-01
The probabilistic solutions of nonlinear stochastic oscillators with even nonlinearity driven by Poisson white noise are investigated in this paper. The stationary probability density function (PDF) of the oscillator responses governed by the reduced Fokker-Planck-Kolmogorov equation is obtained with exponentialpolynomial closure (EPC) method. Different types of nonlinear oscillators are considered. Monte Carlo simulation is conducted to examine the effectiveness and accuracy of the EPC method in this case. It is found that the PDF solutions obtained with EPC agree well with those obtained with Monte Carlo simulation, especially in the tail regions of the PDFs of oscillator responses. Numerical analysis shows that the mean of displacement is nonzero and the PDF of displacement is nonsymmetric about its mean when there is even nonlinearity in displacement in the oscillator. Numerical analysis further shows that the mean of velocity always equals zero and the PDF of velocity is symmetrically distributed about its mean.
On stochastic optimal control of partially observable nonlinear quasi Hamiltonian systems.
Zhu, Wei-qiu; Ying, Zu-guang
2004-11-01
A stochastic optimal control strategy for partially observable nonlinear quasi Hamiltonian systems is proposed. The optimal control forces consist of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimated error of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy. PMID:15495321
A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems
NASA Astrophysics Data System (ADS)
Ying, Z. G.; Zhu, W. Q.
2008-02-01
A stochastic optimal control strategy for partially observable nonlinear quasi-Hamiltonian systems is proposed. The optimal control force consists of two parts. The first part is determined by the conditions under which the stochastic optimal control problem of a partially observable nonlinear system is converted into that of a completely observable linear system. The second part is determined by solving the dynamical programming equation derived by applying the stochastic averaging method and stochastic dynamical programming principle to the completely observable linear control system. The response of the optimally controlled quasi-Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation associated with the optimally controlled completely observable linear system and solving the Riccati equation for the estimate errors of system states. An example is given to illustrate the procedure and effectiveness of the proposed control strategy.
Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems
NASA Astrophysics Data System (ADS)
Feng, Ju; Zhu, Weiqiu; Ying, Zuguang
2010-01-01
The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. The response of the controlled quasi Hamiltonian system is predicted by solving the averaged Fokker-Planck-Kolmogorov equation and the Riccati equation for the estimated error of system states. As an example to illustrate the procedure and effectiveness of the proposed method, the stochastic optimal control problem of a partially observable two-degree-of-freedom quasi-integrable Hamiltonian system is worked out in detail.
External and parametric random excitation of non-linear offshore systems
Thampi, S.K.
1989-01-01
The development of accurate response prediction methods for nonlinear offshore structures is addressed in this study. The Markov approach is adopted for this purpose and the solution methods are illustrated through applications to deepwater offshore systems which include an oceanographic buoy, fixed jacked structures, marine riser systems and a guyed offshore platform. Gaussian and non-Gaussian response predictions for single and multiple degree of freedom systems are presented and discussed at length. The major difficulties associated with Markov methods in dealing with practical systems are the requirements of white noise excitation and the solution of the Fokker-Planck-Kolmogorov equation. These problems are addressed through the development of dimensionless shaping filters to produce realistic excitation and the use of moment equations to compute response statistics. The application of these techniques to non-linear systems requires additional closure approximations. The solutions are compared with those from linear spectral analysis, stochastic averaging and time domain simulations.
On the conditions for the onset of nonlinear chirping structures in NSTX
NASA Astrophysics Data System (ADS)
Duarte, Vinicius; Podesta, Mario; Berk, Herbert; Gorelenkov, Nikolai
2015-11-01
The nonlinear dynamics of phase space structures is a topic of interest in tokamak physics in connection with fast ion loss mechanisms. The onset of phase-space holes and clumps has been theoretically shown to be associated with an explosive solution of an integro-differential, nonlocal cubic equation that governs the early mode amplitude evolution in the weakly nonlinear regime. The existence and stability of the solutions of the cubic equation have been theoretically studied as a function of Fokker-Planck coefficients for the idealized case of a single resonant point of a localized mode. From realistic computations of NSTX mode structures and resonant surfaces, we calculate effective pitch angle scattering and slowing-down (drag) collisional coefficients and analyze NSTX discharges for different cases with respect to chirping experimental observation. Those results are confronted to the theory that predicts the parameters region that allow for chirping to take place.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
A finite element method for the statistics of non-linear random vibration
NASA Astrophysics Data System (ADS)
Langley, R. S.
1985-07-01
The transitional probability density function for the random response of a certain class of non-linear system satisfies the Fokker-Planck-Kolmogorov equation. This paper concerns the numerical solution of the stationary form of this equation, yielding the stationary probability density function of response. The weighted residual statement for the problem is integrated by parts to yield the weak form of the equations, which are then solved by the finite element method. The method is applied to a Duffing oscillator and good agreement is found with the exact result, and the method is compared favourably with a Galerkin solution method given by Bhandari and Sherrer [1]. Also, the method is applied to the ship rolling problem and good agreement is found with an approximate analytical result due to Roberts [2].
NASA Astrophysics Data System (ADS)
Han, Hua; Ding, Yongsheng; Hao, Kuangrong; Hu, Liangjian
2013-07-01
In this article, we first introduce the problem of state estimation of jump Markov nonlinear systems (JMNSs). Since the density evolution method for predictor equations satisfies Fokker-Planck-Kolmogorov equation (FPKE) in Bayes estimation, the FPKE in conjunction with Bayes' conditional density update formula can provide optimal estimation for a general continuous-discrete nonlinear filtering problem. It is well known that the analytical solution of the FPKE and Bayes' formula is extremely difficult to obtain except a few special cases. Hence, we try to design a particle filter to achieve Bayes estimation of the JMNSs. In order to test the viability of our algorithm, we apply it to multiple targets tracking in video surveillance. Before starting simulation, we introduce the 'birth' and 'death' description of targets, targets' transitional probability model, and observation probability. The experiment results show good performance of our proposed filter for multiple targets tracking.
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Linear superposition in nonlinear equations.
Khare, Avinash; Sukhatme, Uday
2002-06-17
Several nonlinear systems such as the Korteweg-de Vries (KdV) and modified KdV equations and lambda phi(4) theory possess periodic traveling wave solutions involving Jacobi elliptic functions. We show that suitable linear combinations of these known periodic solutions yield many additional solutions with different periods and velocities. This linear superposition procedure works by virtue of some remarkable new identities involving elliptic functions. PMID:12059300
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
NASA Astrophysics Data System (ADS)
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Nonlinear SCHRÖDINGER-PAULI Equations
NASA Astrophysics Data System (ADS)
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
NASA Astrophysics Data System (ADS)
Jin, X. L.; Huang, Z. L.
The nonstationary probability densities of system responses are obtained for nonlinear multi-degree-of-freedom systems subject to stochastic parametric and external excitations. First, the stochastic averaging method is used to obtain the averaged Itô equation for amplitude envelopes of the system response. Then, the corresponding Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude envelopes is deduced. By applying the Galerkin method, the nonstationary probability density can be expressed as a series expansion in terms of a set of orthogonal base functions with time-dependent coefficients. Finally, the nonstationary probability densities for the amplitude response, as well as those for the state-space response, are solved approximately. To illustrate the applicability, the proposed method is applied to a two-degree-of-freedom van der Pol oscillator subject to external excitations of Gaussian white noises.
NASA Astrophysics Data System (ADS)
Westerhof, E.; Pratt, J.; Ayten, B.
2015-03-01
In the presence of electron cyclotron current drive (ECCD), the Ohm's law of single fluid magnetohydrodynamics (MHD) is modified as E + v × B = η(J - JECCD). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm's law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands.
Effect of background plasma nonlinearities on dissipation processes in plasmas
NASA Astrophysics Data System (ADS)
Nekrasov, F. M.; Elfimov, A. G.; de Azevedo, C. A.; de Assis, A. S.
1999-01-01
The Coulomb collision effect on the bounce-resonance dissipation is considered for toroidal magnetized plasmas. The solution of the Vlasov equation with a simplified Fokker-Planck collision operator is presented. The parallel components of the dielectric tensor are obtained. A collisionless limit of wave dissipation is found.
NASA Astrophysics Data System (ADS)
Tanimura, Yoshitaka; Steffen, Thomas
2000-12-01
The relaxation processes in a quantum system nonlinearly coupled to a harmonic Gaussian-Markovian heat bath are investigated by the quantum Fokker-Planck equation in the hierarchy form. This model describes frequency fluctuations in the quantum system with an arbitrary correlation time and thus bridges the gap between the Brownian oscillator model and the stochastic model by Anderson and Kubo. The effects of the finite correlation time and the system-bath coupling strength are studied for a harmonic model system by numerically integrating the equation of motion. The one-time correlation function of the system coordinate, which is measured in conventional Raman and infrared absorption experiments, already reflects the inhomogeneous character of the relaxation process. The finite correlation time of the frequency fluctuations, however, is directly evident only in the two- and three-time correlation function as probed by multidimensional spectroscopic techniques such as the Raman echo and the fifth-order 2D Raman experiment.
Vibrational spectroscopy of a harmonic oscillator system nonlinearly coupled to a heat bath
NASA Astrophysics Data System (ADS)
Kato, Tsuyoshi; Tanimura, Yoshitaka
2002-10-01
Vibrational relaxation of a harmonic oscillator nonlinearly coupled to a heat bath is investigated by the Gaussian-Markovian quantum Fokker-Planck equation approach. The system-bath interaction is assumed to be linear in the bath coordinate, but linear plus square in the system coordinate modeling the elastic and inelastic relaxation mechanisms. Interplay of the two relaxation processes induced by the linear-linear and square-linear interactions in Raman or infrared spectra is discussed for various system-bath couplings, temperatures, and correlation times for the bath fluctuations. The one-quantum coherence state created through the interaction with the pump laser pulse relaxes through different pathways in accordance with the mechanisms of the system-bath interactions. Relations between the present theory, Redfield theory, and stochastic theory are also discussed.
An exact solution to a certain non-linear random vibration problem
NASA Astrophysics Data System (ADS)
Dimentberg, M. F.
A single-degree-of-freedom system with a special type of nonlinear damping and both external and parametric white-noise excitations is considered. For the special case, when the intensities of coordinates and velocity modulation satisfy a certain condition an exact analytical solution is obtained to the corresponding stationary Fokker-Planck-Kolmogorov equation yielding an expression for joint probability density of coordinate and velocity. This solution is analyzed particularly in connection with stochastic stability problem for the corresponding linear system; certain implications are illustrated for the system, which is stable with respect to probability but unstable in the mean square. The solution obtained may be used to check different approximate methods for analysis of systems with randomly varying parameters.
Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states
2011-01-01
Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter. AMS Subject Classification: 35K60, 82C31, 92B20. PMID:22657097
NASA Astrophysics Data System (ADS)
Shinozaki, Takashi; Okada, Masato; Reyes, Alex D.; Câteau, Hideyuki
2010-01-01
Intermingled neural connections apparent in the brain make us wonder what controls the traffic of propagating activity in the brain to secure signal transmission without harmful crosstalk. Here, we reveal that inhibitory input but not excitatory input works as a particularly useful traffic controller because it controls the degree of synchrony of population firing of neurons as well as controlling the size of the population firing bidirectionally. Our dynamical system analysis reveals that the synchrony enhancement depends crucially on the nonlinear membrane potential dynamics and a hidden slow dynamical variable. Our electrophysiological study with rodent slice preparations show that the phenomenon happens in real neurons. Furthermore, our analysis with the Fokker-Planck equations demonstrates the phenomenon in a semianalytical manner.
Harvey, R. W.; Petrov, Yu. V.
2013-12-03
Within the US Department of Energy/Office of Fusion Energy magnetic fusion research program, there is an important whole-plasma-modeling need for a radio-frequency/neutral-beam-injection (RF/NBI) transport-oriented finite-difference Fokker-Planck (FP) code with combined capabilities for 4D (2R2V) geometry near the fusion plasma periphery, and computationally less demanding 3D (1R2V) bounce-averaged capabilities for plasma in the core of fusion devices. Demonstration of proof-of-principle achievement of this goal has been carried out in research carried out under Phase I of the SBIR award. Two DOE-sponsored codes, the CQL3D bounce-average Fokker-Planck code in which CompX has specialized, and the COGENT 4D, plasma edge-oriented Fokker-Planck code which has been constructed by Lawrence Livermore National Laboratory and Lawrence Berkeley Laboratory scientists, where coupled. Coupling was achieved by using CQL3D calculated velocity distributions including an energetic tail resulting from NBI, as boundary conditions for the COGENT code over the two-dimensional velocity space on a spatial interface (flux) surface at a given radius near the plasma periphery. The finite-orbit-width fast ions from the CQL3D distributions penetrated into the peripheral plasma modeled by the COGENT code. This combined code demonstrates the feasibility of the proposed 3D/4D code. By combining these codes, the greatest computational efficiency is achieved subject to present modeling needs in toroidally symmetric magnetic fusion devices. The more efficient 3D code can be used in its regions of applicability, coupled to the more computationally demanding 4D code in higher collisionality edge plasma regions where that extended capability is necessary for accurate representation of the plasma. More efficient code leads to greater use and utility of the model. An ancillary aim of the project is to make the combined 3D/4D code user friendly. Achievement of full-coupling of these two Fokker-Planck
NASA Astrophysics Data System (ADS)
Frank, T. D.
2007-01-01
We present a generalized Kramers Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker Planck equation derived earlier in the literature is a special case of the proposed Kramers Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed.
Generalization of the diffusion equation by using the maximum entropy principle
NASA Astrophysics Data System (ADS)
Jumarie, Guy
1985-06-01
By using the so-called maximum entropy principle in information theory, one derives a generalization of the Fokker-Planck-Kolmogorov equation which applies when the n first transition moments of the process are proportional to Δt, while the other ones can be neglected.
Engineering integrable nonautonomous nonlinear Schroedinger equations
He Xugang; Zhao Dun; Li Lin; Luo Honggang
2009-05-15
We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schroedinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
Stochastic inflation and nonlinear gravity
NASA Astrophysics Data System (ADS)
Salopek, D. S.; Bond, J. R.
1991-02-01
We show how nonlinear effects of the metric and scalar fields may be included in stochastic inflation. Our formalism can be applied to non-Gaussian fluctuation models for galaxy formation. Fluctuations with wavelengths larger than the horizon length are governed by a network of Langevin equations for the physical fields. Stochastic noise terms arise from quantum fluctuations that are assumed to become classical at horizon crossing and that then contribute to the background. Using Hamilton-Jacobi methods, we solve the Arnowitt-Deser-Misner constraint equations which allows us to separate the growing modes from the decaying ones in the drift phase following each stochastic impulse. We argue that the most reasonable choice of time hypersurfaces for the Langevin system during inflation is T=ln(Ha), where H and a are the local values of the Hubble parameter and the scale factor, since T is the natural time for evolving the short-wavelength scalar field fluctuations in an inhomogeneous background. We derive a Fokker-Planck equation which describes how the probability distribution of scalar field values at a given spatial point evolves in T. Analytic Green's-function solutions obtained for a single scalar field self-interacting through an exponential potential are used to demonstrate (1) if the initial condition of the Hubble parameter is chosen to be consistent with microwave-background limits, H(φ0)/mρ<~10-4, then the fluctuations obey Gaussian statistics to a high precision, independent of the time hypersurface choice and operator-ordering ambiguities in the Fokker-Planck equation, and (2) for scales much larger than our present observable patch of the Universe, the distribution is non-Gaussian, with a tail extending to large energy densities; although there are no observable manifestations, it does show eternal inflation. Lattice simulations of our Langevin network for the exponential potential demonstrate how spatial correlations are incorporated. An initially
Spurious Solutions Of Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Yee, H. C.; Sweby, P. K.; Griffiths, D. F.
1992-01-01
Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.
Numerical methods for high-dimensional probability density function equations
NASA Astrophysics Data System (ADS)
Cho, H.; Venturi, D.; Karniadakis, G. E.
2016-01-01
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
Nonlinear quantum equations: Classical field theory
Rego-Monteiro, M. A.; Nobre, F. D.
2013-10-15
An exact classical field theory for nonlinear quantum equations is presented herein. It has been applied recently to a nonlinear Schrödinger equation, and it is shown herein to hold also for a nonlinear generalization of the Klein-Gordon equation. These generalizations were carried by introducing nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard, linear equations, are recovered in the limit q→ 1. The main characteristic of this field theory consists on the fact that besides the usual Ψ(x(vector sign),t), a new field Φ(x(vector sign),t) needs to be introduced in the Lagrangian, as well. The field Φ(x(vector sign),t), which is defined by means of an additional equation, becomes Ψ{sup *}(x(vector sign),t) only when q→ 1. The solutions for the fields Ψ(x(vector sign),t) and Φ(x(vector sign),t) are found herein, being expressed in terms of a q-plane wave; moreover, both field equations lead to the relation E{sup 2}=p{sup 2}c{sup 2}+m{sup 2}c{sup 4}, for all values of q. The fact that such a classical field theory works well for two very distinct nonlinear quantum equations, namely, the Schrödinger and Klein-Gordon ones, suggests that this procedure should be appropriate for a wider class nonlinear equations. It is shown that the standard global gauge invariance is broken as a consequence of the nonlinearity.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
The quasicontinuum Fokker-Plank equation
Alexander, Francis J
2008-01-01
We present a regularized Fokker-Planck equation with more accurate short-time and high-frequency behavior for continuous-time, discrete-state systems. The regularization preserves crucial aspects of state-space discreteness lost in the standard Kramers-Moyal expansion. We apply the method to a simple example of biochemical reaction kinetics and to a two-dimensional symmetric random walk, and suggest its application to more complex systerns.
Markovian master equation for nonlinear systems
NASA Astrophysics Data System (ADS)
de los Santos-Sánchez, O.; Récamier, J.; Jáuregui, R.
2015-06-01
Within the f-deformed oscillator formalism, we derive a Markovian master equation for the description of the damped dynamics of nonlinear systems that interact with their environment. The applicability of this treatment to the particular case of a Morse-like oscillator interacting with a thermal field is illustrated, and the decay of quantum coherence in such a system is analyzed in terms of the evolution on phase space of its nonlinear coherent states via the Wigner function.
A Stochastic Differential Equation Approach To Multiphase Flow In Porous Media
NASA Astrophysics Data System (ADS)
Dean, D.; Russell, T.
2003-12-01
The motivation for using stochastic differential equations in multiphase flow systems stems from our work in developing an upscaling methodology for single phase flow. The long term goals of this project include: I. Extending this work to a nonlinear upscaling methodology II. Developing a macro-scale stochastic theory of multiphase flow and transport that accounts for micro-scale heterogeneities and interfaces. In this talk, we present a stochastic differential equation approach to multiphase flow, a typical example of which is flow in the unsaturated domain. Specifically, a two phase problem is studied which consists of a wetting phase and a non-wetting phase. The approach given results in a nonlinear stochastic differential equation describing the position of the non-wetting phase fluid particle. Our fundamental assumption is that the flow of fluid particles is described by a stochastic process and that the positions of the fluid particles over time are governed by the law of the process. It is this law which we seek to determine. The nonlinearity in the stochastic differential equation arises because both the drift and diffusion coefficients depend on the volumetric fraction of the phase which in turn depends on the position of the fluid particles in the experimental domain. The concept of a fluid particle is central to the development of the model described in this talk. Expressions for both saturation and volumetric fraction are developed using the fluid particle concept. Darcy's law and the continuity equation are then used to derive a Fokker-Planck equation using these expressions. The Ito calculus is then applied to derive a stochastic differential equation for the non-wetting phase. This equation has both drift and diffusion terms which depend on the volumetric fraction of the non-wetting phase. Standard stochastic theories based on the Ito calculus and the Wiener process and the equivalent Fokker-Planck PDE's are typically used to model dispersion
Algorithms For Integrating Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
Numerical Methods for Stochastic Partial Differential Equations
Sharp, D.H.; Habib, S.; Mineev, M.B.
1999-07-08
This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
NASA Astrophysics Data System (ADS)
Thomson, Mark J.; McKellar, Bruce H. J.
1991-04-01
A simple, non-linear generalization of the MSW equation is presented and its analytic solution is outlined. The orbits of the polarization vector are shown to be periodic, and to lie on a sphere. Their non-trivial flow patterns fall into two topological categories, the more complex of which can become chaotic if perturbed.
Quantum nonlinear Schrodinger equation on a lattice
Bogolyubov, N.M.; Korepin, V.E.
1986-09-01
A local Hamiltonian is constructed for the nonlinear Schrodinger equation on a lattice in both the classical and the quantum variants. This Hamiltonian is an explicit elementary function of the local Bose fields. The lattice model possesses the same structure of the action-angle variables as the continuous model.
Two coupled nonlinear cavities in a driven-dissipative environment
NASA Astrophysics Data System (ADS)
Cao, Bin; Mahmud, Khan; Hafezi, Mohammad
We investigate two coupled nonlinear cavities that are driven coherently in a dissipative environment. This is the simplest setting containing a good number of features of an array of coupled cavity quantum simulator with Kerr nonlinearity which gives rise to many strongly correlated phases. We find analytical solution for the steady state using the generalized P representation and expressing the master equation in the form of Fokker-Planck equation. A comparison shows a good match of the analytical and numerical solutions across different regimes. We investigate the quantum correlations in the steady state by solving the full master equation numerically, analyzing its second-order coherence, entanglment entropy and Liouvillian gap as a function of drive and detuning. This gives us insights into the nature of bistability and how the tunneling-induced bistability emerges in coupled cavities when going beyond a single cavity. We can understand much of the semiclassical physics in terms of the underlying phase space dynamics of a driven and damped classical pendulum. Furthermore, in the semiclassical analysis, we find steady state solutions with different number density in the two wells that can be considered an analog of double well self-trapped states.
Non-equilibrium effects upon the non-Markovian Caldeira-Leggett quantum master equation
Bolivar, A.O.
2011-05-15
Highlights: > Classical Brownian motion described by a non-Markovian Fokker-Planck equation. > Quantization process. > Quantum Brownian motion described by a non-Markovian Caldeira-Leggett equation. > A non-equilibrium quantum thermal force is predicted. - Abstract: We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker-Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira-Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.
Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Mihaila, Bogdan; Saxena, Avadh
2010-09-01
We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction g{2}/k+1(ΨΨ){k+1} , as well as a vector-vector self interaction g{2}/k+1(Ψγ{μ}ΨΨγ{μ}Ψ){1/2(k+1)} . We find the exact analytic form for solitary waves for arbitrary k and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the 1/2m correction to the NLSE, valid when |ω-m|<2m , where ω is the frequency of the solitary wave in the rest frame. We discuss the stability and blowup of solitary waves assuming the modified NLSE is valid and find that they should be stable for k<2 . PMID:21230200
Taming the nonlinearity of the Einstein equation.
Harte, Abraham I
2014-12-31
Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically interesting cases where metrics become approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations that directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well. PMID:25615299
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
NASA Technical Reports Server (NTRS)
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
Forces Associated with Nonlinear Nonholonomic Constraint Equations
NASA Technical Reports Server (NTRS)
Roithmayr, Carlos M.; Hodges, Dewey H.
2010-01-01
A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications.
Transport equations for low-energy solar particles in evolving interplanetary magnetic fields
NASA Technical Reports Server (NTRS)
Ng, C. K.
1988-01-01
Two new forms of a simplified Fokker-Planck equation are derived for the transport of low-energy solar energetic particles in an evolving interplanetary magnetic field, carried by a variable radial solar wind. An idealized solution suggests that the 'invariant' anisotropy direction reported by Allum et al. (1974) may be explained within the conventional theoretical framework. The equations may be used to relate studies of solar particle propagation to solar wind transients, and vice versa.
Nonlocal growth equations-a test case for dynamic renormalization group analysis
NASA Astrophysics Data System (ADS)
Schwartz, Moshe; Katzav, Eytan
2003-12-01
In this paper we discuss nonlocal growth equations such as the generalization of the Kardar-Parisi-Zhang (KPZ) equation that includes long-range interactions, also known as the Nonlocal-Kardar-Parisi-Zhang (NKPZ) equation, and the nonlocal version of the molecular-beam-epitaxy (NMBE) equation. We show that the steady-state strong coupling solution for nonlocal models such as NKPZ and NMBE can be obtained exactly in one dimension for some special cases, using the Fokker-Planck form of these equations. The exact results we derive do not agree with previous results obtained by Dynamic Renormalization Group (DRG) analysis. This discrepancy is important because DRG is a common method used extensively to deal with nonlinear field equations. While difficulties with this method for d>1 has been realized in the past, it has been believed so far that DRG is still safe in one dimension. Our result shows differently. The reasons for the failure of DRG to recover the exact one-dimensional results are also discussed.
Dark soliton solutions of (N+1)-dimensional nonlinear evolution equations
NASA Astrophysics Data System (ADS)
Demiray, Seyma Tuluce; Bulut, Hasan
2016-06-01
In this study, we investigate exact solutions of (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation by using generalized Kudryashov method. (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation can be returned to nonlinear ordinary differential equation by suitable transformation. Then, generalized Kudryashov method has been used to seek exact solutions of the (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation. Also, we obtain dark soliton solutions for these (N+1)-dimensional nonlinear evolution equations. Finally, we denote that this method can be applied to solve other nonlinear evolution equations.
Using the scalable nonlinear equations solvers package
Gropp, W.D.; McInnes, L.C.; Smith, B.F.
1995-02-01
SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of large-scale systems of nonlinear equations on both uniprocessors and parallel architectures. SNES also contains a component for the solution of unconstrained minimization problems, called SUMS (Scalable Unconstrained Minimization Solvers). Newton-like methods, which are known for their efficiency and robustness, constitute the core of the package. As part of the multilevel PETSc library, SNES incorporates many features and options from other parts of PETSc. In keeping with the spirit of the PETSc library, the nonlinear solution routines are data-structure-neutral, making them flexible and easily extensible. This users guide contains a detailed description of uniprocessor usage of SNES, with some added comments regarding multiprocessor usage. At this time the parallel version is undergoing refinement and extension, as we work toward a common interface for the uniprocessor and parallel cases. Thus, forthcoming versions of the software will contain additional features, and changes to parallel interface may result at any time. The new parallel version will employ the MPI (Message Passing Interface) standard for interprocessor communication. Since most of these details will be hidden, users will need to perform only minimal message-passing programming.
The beam equation with nonlinear memory
NASA Astrophysics Data System (ADS)
D'Abbicco, Marcello; Lucente, Sandra
2016-06-01
In this paper, we study the critical exponent for the beam equation with nonlinear memory, i.e., {u_{tt}+Δ^2u = F(t, u)}, where F = intlimits0tf(t - s)N(u)(s, x) {d}s, quad N(u)≈ |u|^p. For suitable f and p, we prove the existence of local-in-time solutions and small data global solutions to the Cauchy problem, in homogeneous and nonhomogeneous Sobolev spaces. In some cases, we prove that the local solution cannot be extended to a global one. We also consider the limit case of power nonlinearity, i.e., {F = N(u)}.
NASA Astrophysics Data System (ADS)
Er, Guo-Kang
2014-04-01
In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker-Planck-Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler-Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.
Exact and explicit solitary wave solutions to some nonlinear equations
Jiefang Zhang
1996-08-01
Exact and explicit solitary wave solutions are obtained for some physically interesting nonlinear evolutions and wave equations in physics and other fields by using a special transformation. These equations include the KdV-Burgers equation, the MKdV-Burgers equation, the combined KdV-MKdV equation, the Newell-Whitehead equation, the dissipative {Phi}{sup 4}-model equation, the generalized Fisher equation, and the elastic-medium wave equation.
FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations
NASA Astrophysics Data System (ADS)
Ibragimov, N. H.; Torrisi, M.; Tracinà, R.
2010-11-01
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.
Solution spectrum of nonlinear diffusion equations
Ulmer, W.
1992-08-01
The stationary version of the nonlinear diffusion equation -{partial_derivative}c/{partial_derivative}t+D{Delta}c=A{sub 1}c-A{sub 2}c{sup 2} can be solved with the ansatz c={summation}{sub p=1}{sup {infinity}} A{sub p}(cosh kx){sup -p}, inducing a band structure with regard to the ratio {lambda}{sub 1}/{lambda}{sub 2}. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c={summation}{sub p,q=1}{sup infinity}A{sub pa}(cosh kx){sup -p}[(cosh {alpha}t){sup -q-1} sinh {alpha}t+b(cosh {alpha}t){sup -q}] represents a solution spectrum of the time-dependent nonlinear equations, and the stationary version can be found from the asymptotic behaviour of the expansion. The solutions can be associated with reactive processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheriods. 33 refs., 1 tab.
Nonlinear scattering term in the gyrokinetic Vlasov equation
Wang, Shaojie
2013-08-15
Nonlinear scattering term is found from the nonlinear gyrokinetic equation by decoupling the perturbed gyrocenter motion from the unperturbed motion. The gyro-center distribution function is determined by the well-understood unperturbed motion, with the effects of fields perturbation included in the nonlinear scattering term, which explicitly reveals the nonlinear stochastic dissipation on the time scale longer than the wave correlation time.
Westerhof, E. Pratt, J.
2014-10-15
In the presence of electron cyclotron current drive (ECCD), the Ohm's law of single fluid magnetohydrodynamics is modified as E + v × B = η(J – J{sub EC}). This paper presents a new closure relation for the EC driven current density appearing in this modified Ohm's law. The new relation faithfully represents the nonlocal character of the EC driven current and its main origin in the Fisch-Boozer effect. The closure relation is validated on both an analytical solution of an approximated Fokker-Planck equation as well as on full bounce-averaged, quasi-linear Fokker-Planck code simulations of ECCD inside rotating magnetic islands. The new model contains the model put forward by Giruzzi et al. [Nucl. Fusion 39, 107 (1999)] in one of its limits.
Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation.
Yan, Zhenya
2013-04-28
The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report on exact solutions of one- and two-dimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex -symmetric potentials. Finally, some complex -symmetric extension principles are used to generate some complex -symmetric nonlinear wave equations starting from both -symmetric (e.g. the KdV equation) and non- -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex -symmetric Burgers equation in detail. PMID:23509385
Forced nonlinear Schrödinger equation with arbitrary nonlinearity
NASA Astrophysics Data System (ADS)
Cooper, Fred; Khare, Avinash; Quintero, Niurka R.; Mertens, Franz G.; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction (g2)/(κ+1)(ψψ)κ+1 in the presence of the external forcing terms of the form re-i(kx+θ)-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where vk=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq˙(t)<0, where p(t) is the normalized canonical momentum p(t)=(1)/(M(t))(∂L)/(∂q˙), and q˙(t) is the solitary wave velocity. Here M(t)=∫dxψ(x,t)ψ(x,t). Stability is also studied using a “phase portrait” of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.
Castellano, Claudio; Muñoz, Miguel A; Pastor-Satorras, Romualdo
2009-10-01
We introduce a nonlinear variant of the voter model, the q-voter model, in which q neighbors (with possible repetition) are consulted for a voter to change opinion. If the q neighbors agree, the voter takes their opinion; if they do not have a unanimous opinion, still a voter can flip its state with probability epsilon . We solve the model on a fully connected network (i.e., in mean field) and compute the exit probability as well as the average time to reach consensus by employing the backward Fokker-Planck formalism and scaling arguments. We analyze the results in the perspective of a recently proposed Langevin equation aimed at describing generic phase transitions in systems with two ( Z2-symmetric) absorbing states. In particular, by deriving explicitly the coefficients of such a Langevin equation as a function of the microscopic flipping probabilities, we find that in mean field the q-voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other: (i) a unique (generalized-voter-like) transition, (ii) a series of two consecutive transitions, one (Ising-like) in which the Z2 symmetry is broken and a separate one (in the directed-percolation class) in which the system falls into an absorbing state, and (iii) a series of two transitions, including an intermediate regime in which the final state depends on initial conditions. This third (so far unexplored) scenario, in which a type of ordering dynamics emerges, is rationalized and found to be specific of mean field, i.e., fluctuations are explicitly shown to wash it out in spatially extended systems. PMID:19905295
Capillary waves in the subcritical nonlinear Schroedinger equation
Kozyreff, G.
2010-01-15
We expand recent results on the nonlinear Schroedinger equation with cubic-quintic nonlinearity to show that some solutions are described by the Bernoulli equation in the presence of surface tension. As a consequence, capillary waves are predicted and found numerically at the interface between regions of large and low amplitude.
Forced nonlinear Schrödinger equation with arbitrary nonlinearity.
Cooper, Fred; Khare, Avinash; Quintero, Niurka R; Mertens, Franz G; Saxena, Avadh
2012-04-01
We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq ̇(t)<0, where p(t) is the normalized canonical momentum p(t)=1/M(t)∂L/∂q ̇, and q ̇(t) is the solitary wave velocity. Here M(t)=∫dxψ*(x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave. PMID:22680598
Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method
NASA Astrophysics Data System (ADS)
Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet
2015-10-01
The auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis. PMID:27347461
Stochastic differential equations for non-linear hydrodynamics
NASA Astrophysics Data System (ADS)
Español, Pep
1998-02-01
We formulate the stochastic differential equations for non-linear hydrodynamic fluctuations. The equations incorporate the random forces through a random stres tensor and random heat flux as in the Landau and Lifshitz theory. However, the equations are non-linear and the random forces are non-Gaussian. We provide explicit expressions for these random quantities in terms of the well-defined increments of the Wienner process.
Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation
Ismail, M. S.
2010-09-30
The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.
Analytic solutions of a general nonlinear functional equations near resonance
NASA Astrophysics Data System (ADS)
Xu, Bing; Zhang, Weinian
2006-05-01
Existence of analytic solutions of a general class of nonlinear functional equations is discussed. This general class includes some specific functional equations studied recently. Moreover, we can generalize this problem to finding analytic solutions of a general class of iterative equations.
The zero dispersion limits of nonlinear wave equations
Tso, T.
1992-01-01
In chapter 2 the author uses functional analytic methods and conservation laws to solve the initial-value problem for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation, and the nonlinear Schroedinger equation for initial data that satisfy some suitable conditions. In chapter 3 the energy estimates are used to show that the strong convergence of the family of the solutions of the KdV equation obtained in chapter 2 in H[sup 3](R) as [epsilon] [yields] 0; also, it is shown that the strong L[sup 2](R)-limit of the solutions of the BBM equation as [epsilon] [yields] 0 before a critical time. In chapter 4 the author uses the Whitham modulation theory and averaging method to find the 2[pi]-periodic solutions and the modulation equations of the KdV equation, the BBM equation, the Klein-Gordon equation, the NLS equation, the mKdV equation, and the P-system. It is shown that the modulation equations of the KdV equation, the K-G equation, the NLS equation, and the mKdV equation are hyperbolic but those of the BBM equation and the P-system are not hyperbolic. Also, the relations are studied of the KdV equation and the mKdV equation. Finally, the author studies the complex mKdV equation to compare with the NLS equation, and then study the complex gKdV equation.
NASA Astrophysics Data System (ADS)
Kato, Tsuyoshi; Tanimura, Yoshitaka
2004-01-01
Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.