Stability analysis of implicit time discretizations for the Compton-scattering Fokker–Planck equation

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.     

Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed.     

Stochastic feedback,nonlinear families of Markov processes,and nonlinear Fokker–Planck equations

We discuss two fundamental aspects of Fokker–Planck equations that are nonlinear with respect to probability densities. First, we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions. Second, we demonstrate that nonlinear Fokker–Planck equations can be interpreted as linear Fokker–Planck equations describing nonlinear families of Markov diffusion processes. We exploit this finding in order to derive complete hierarchies of probability densities from nonlinear Fokker–Planck equations.     

A nonlinear Fokker–Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

We investigate the solutions for a set of coupled nonlinear Fokker–Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.     

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.     

A Vlasov–Fokker–Planck code for high energy density physics

OSHUN is a parallel relativistic 2D3P Vlasov–Fokker–Planck code, developed primarily to study electron transport and instabilities pertaining to laser-produced—including laser-fusion—plasmas. It incorporates a spherical harmonic expansion of the electron distribution function, where the number of terms is an input parameter that determines the angular resolution in momentum-space. The algorithm employs the full 3D electromagnetic fields and a rigorous linearized Fokker–Planck collision operator. The numerical scheme conserves energy and number density. This enables simulations for plasmas with temperatures from MeV down to a few eV and densities from less than critical to more than solid. Kinetic phenomena as well as electron transport physics can be recovered accurately and efficiently.     

Fokker–Planck Theory of Nonequilibrium Systems Governed by Hierarchical Dynamics

Dynamics of complex systems is often hierarchically organized on different time scales. To understand the physics of such hierarchy, here Brownian motion of a particle moving through a fluctuating medium with slowly varying temperature is studied as an analytically tractable example, and a kinetic theory is formulated for describing the states of the particle. What is peculiar here is that the (inverse) temperature is treated as a dynamical variable. Dynamical hierarchy is introduced in conformity with the adiabatic scheme. Then, a new analytical method is developed to show how the Fokker–Planck equation admits as a stationary solution the Maxwellian distribution modulated by the temperature fluctuations, the distribution of which turns out to be determined by the drift term. A careful comment is also made on so-called superstatistics.     

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.     

A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma

The interaction of intense lasers with solid matter generates a hot plasma state that is well described by the Vlasov–Fokker–Planck equation. Accurate and efficient modeling of the physics in these scenarios is highly pertinent, because it relates to experimental campaigns to produce energy by inertial confinement fusion on facilities such as the National Ignition Facility. Calculations involving the Vlasov–Fokker–Planck equation are computationally intensive, but are crucial to proper understanding of a wide variety of physical effects and instabilities in inertial fusion plasmas. In this topical review, we will introduce the background physics related to Vlasov–Fokker–Planck simulation, and then proceed to describe results from numerical simulation of inertial fusion plasma in a pedagogical manner by discussing some key numerical algorithm developments that enabled the research to take place. A qualitative comparison of the techniques is also given.     

A mixed SOC-turbulence model for nonlocal transport and Lévy-fractional Fokker–Planck equation

The phenomena of nonlocal transport in magnetically confined plasma are theoretically analyzed. A hybrid model is proposed, which brings together the notion of inverse energy cascade, typical of drift-wave- and two-dimensional fluid turbulence, and the ideas of avalanching behavior, associable with self-organized critical (SOC) behavior. Using statistical arguments, it is shown that an amplification mechanism is needed to introduce nonlocality into dynamics. We obtain a consistent derivation of nonlocal Fokker–Planck equation with space-fractional derivatives from a stochastic Markov process with the transition probabilities defined in reciprocal space. The hybrid model observes the Sparre Andersen universality and defines a new universality class of SOC.     

Numerical approximation of the Vlasov–Maxwell–Fokker–Planck system in two dimensions

A numerical method is developed for solving the Vlasov–Maxwell–Fokker–Planck system in two spatial dimensions. This system of equations is a model for a collisional plasma in the presence of a self consistent electromagnetic field. The numerical procedure is a type of deterministic particle method and is an extension to include the full electromagnetic field of the approximation method of Wollman and Ozizmir [S. Wollman, E. Ozizmir, Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions, J. Comput. Phys. 228 (2009) 6629–6669]. In addition, the long time asymptotic behavior of solutions is studied. It is determined that the solution to the Vlasov–Maxwell–Fokker–Planck system converges to the same steady state solution as that for the Vlasov–Poisson–Fokker–Planck system.     

Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

In this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial-boundary value problems for a class of limited diffusion Fokker–Planck equations under homogeneous Neumann boundary conditions. We show stability and positivity preserving property under a Courant–Friedrichs–Lewy parabolic time step restriction. We focus on the relativistic heat equation as a model problem of the mentioned limited diffusion Fokker–Planck equations. We analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport. We present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time. We perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi-implicit nonlinear Crank–Nicolson type scheme.     

Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions

A numerical method is developed for approximating the solution to the Vlasov–Poisson–Fokker–Planck system in two spatial dimensions. The method generalizes the approximation for the system in one dimension given in [S. Wollman, E. Ozizmir, Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. The numerical procedure is based on a change of variables that puts the convection–diffusion equation into a form so that finite difference methods for parabolic type partial differential equations can be applied. The computational cycle combines a type of deterministic particle method with a periodic interpolation of the solution along particle trajectories onto a fixed grid. computational work is done to demonstrate the accuracy and effectiveness of the approximation method. Parts of the numerical procedure are adapted to run on a parallel computer.