Fokker–Planck Equation for a Metastable Time Dependent Potential

We use the Fokker–Planck equation to study the diffusion process driven for a metastable potential within a temporal dependence. This potential is characterized by the existence of a barrier that increases with time and reduces the particle diffusion. Escape rate across the barrier for different values of diffusion coefficient is analyzed. The results are also associated with the diffusion process through ion channels in biological system.     

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.     

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.     

A class of asymptotic-preserving schemes for the Fokker–Planck–Landau equation

We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker–Planck operator that can be inverted by the conjugate-gradient method. We compare the effects of different penalization operators, and conclude that the Fokker–Planck (FP) operator is a good choice. Such schemes overcome the stiffness of the collision operator in the fluid regime, and can capture the fluid dynamic limit without numerically resolving the small Knudsen number. Numerical experiments demonstrate that the schemes possess the AP property for general initial data, with numerical accuracy uniformly in the Knudsen number.     

Solution to the Fokker–Planck Equation with Piecewise-Constant Drift

We study the solution to the Fokker–Planck equation with piecewise-constant drift, taking the case with two jumps in the drift as an example. The solution in Laplace space can be expressed in closed analytic form, and its inverse can be obtained conveniently using some numerical inversion methods. The results obtained by numerical inversion can be regarded as exact solutions, enabling us to demonstrate the validity of some numerical methods for solving the Fokker–Planck equation. In particular, ...     

The Linearized Vlasov and Vlasov–Fokker–Planck Equations in a Uniform Magnetic Field

Journal of Statistical Physics - We study the linearized Vlasov equations and the linearized Vlasov–Fokker–Planck equations in the weakly collisional limit in a uniform magnetic field....     

Conditional velocity statistics in the double scalar mixing layer – A mapping closure approach

In this work we use 3D direct numerical simulations (DNS) to investigate the average velocity conditioned on a conserved scalar in a double scalar mixing layer (DSML). The DSML is a canonical multistream flow designed as a model problem for the extensively studied piloted diffusion flames. The conditional mean velocity appears as an unclosed term in advanced Eulerian models of turbulent non-premixed combustion, like the conditional moment closure and transported probability density function (PDF) methods. Here it accounts for inhomogeneous effects that have been found significant in flames with relatively low Damköhler numbers. Today there are only a few simple models available for the conditional mean velocity and these are discussed with reference to the DNS results. We find that both the linear model of Kutznetzov and the Li and Bilger model are unsuitable for multi stream flows, whereas the gradient diffusion model of Pope shows very close agreement with DNS over the whole range of the DSML. The gradient diffusion model relies on a model for the conserved scalar PDF and here we have used a presumed mapping function PDF, that is known to give an excellent representation of the DNS. A new model for the conditional mean velocity is suggested by arguing that the Gaussian reference field represents the velocity field, a statement that is evidenced by a near perfect agreement with DNS. The model still suffers from an inconsistency with the unconditional flux of conserved scalar variance, though, and a strategy for developing fully consistent models is suggested.     

Fick and Fokker–Planck Diffusion Law in Inhomogeneous Media

Journal of Statistical Physics - We discuss particle diffusion in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice....     

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.     

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.     

A New Derivation of the Quantum Navier–Stokes Equations in the Wigner–Fokker–Planck Approach

A quantum Navier–Stokes system for the particle, momentum, and energy densities is formally derived from the Wigner–Fokker–Planck equation using a moment method. The viscosity term depends on the particle density with a shear viscosity coefficient which equals the quantum diffusion coefficient of the Fokker–Planck collision operator. The main idea of the derivation is the use of a so-called osmotic momentum operator, which is the sum of the phase-space momentum and the gradient operator. In this way, a Chapman–Enskog expansion of the Wigner function, which typically leads to viscous approximations, is avoided. Moreover, we show that the osmotic momentum emerges from local gauge theory.     

A Rigorous Derivation of a Linear Kinetic Equation of Fokker–Planck Type in the Limit of Grazing Collisions

We rigorously derive a linear kinetic equation of Fokker–Planck type for a 2-D Lorentz gas in which the obstacles are randomly distributed. Each obstacle of the Lorentz gas generates a potential V( ), where V is a smooth radially symmetric function with compact support, and >0. The density of obstacles diverges as ^– , where >0. We prove that when 0< <1/8 and =2+1, the probability density of a test particle converges as 0 to a solution of our kinetic equation.     

On the Derivation of a High-Velocity Tail from the Boltzmann–Fokker–Planck Equation for Shear Flow

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 Vv–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()lim ₀ ^–2 (–), where 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 _th0.3520 (where is an average collision frequency and a _th/ is the real root of the cubic equation 64x ³+16x ²+12x–9=0) the velocity distribution function exhibits an algebraic high-velocity tail of the form f(V;a)|V|^–4–(a) (;a), where tan V _y /V _x and the angular distribution function (;a) is the solution of a modified Mathieu equation. The enforcement of the periodicity condition (;a)=(+;a) allows one to obtain the exponent (a) as a function of the shear rate. It diverges when aa _th and tends to a minimum value _min1.252 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+(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 ~(a), which rotates from ~=–/4,3/4 when aa _th to ~=0, in the limit a.     

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.     

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases

We propose an extension of the Fokker–Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier–Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar–Gross–Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman–Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.