Solving the Fokker-Planck kinetic equation on a lattice

TL;DR

This paper introduces a lattice-based numerical scheme for solving the Fokker-Planck kinetic equation in multiple dimensions, extending previous one-dimensional models and demonstrating its stability and efficiency through simulations.

Contribution

It develops a generalized Hermite-Gauss discretization and a Chapman-Enskog expansion for the lattice Fokker-Planck equation, enabling accurate macroscopic reproductions in higher dimensions.

Findings

The scheme reproduces continuum equations accurately.

The algorithm is stable with respect to time-step variations.

Preliminary simulations show efficiency in modeling Brownian particles.

Abstract

We propose a discrete lattice version of the Fokker-Planck kinetic equation along lines similar to the Lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension $D$. A generalized Hermite-Gauss procedure is used to construct a discretized kinetic equation and a Chapman-Enskog expansion is applied to adapt the scheme so as to correctly reproduce the macroscopic continuum equations. The stability of the algorithm with respect to the finite time-step $\Delta t$ is characterized by the eigenvalues of the collision matrix. A heuristic second-order algorithm in $\Delta t$ is applied to investigate the time evolution of the distribution function of simple model systems, and compared to known analytical solutions. Preliminary investigations of sedimenting Brownian particles subjected to an orthogonal centrifugal force illustrate the numerical efficiency of the Lattice-Fokker-Planck algorithm to simulate non-trivial situations. Interactions between Brownian particles may be accounted for by adding a standard BGK collision operator to the discretized Fokker-Planck kernel.