Random Walk Approximation for Irreversible Drift-Diffusion Process on Manifold: Ergodicity, Unconditional Stability and Convergence (Communications in Computational Physics, 2023)
Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which doesnot satisfy detailed balance. For the corresponding Fokker-Planck equation on a closedmanifold, using Voronoi tessellation, we propose two upwind finite volume schemeswith or without the information of the invariant measure. Both schemes possessstochastic $Q$-matrix structures and can be decomposed as a gradient flow part anda Hamiltonian flow part, enabling us to prove unconditional stability, ergodicity anderror estimates. Based on the two upwind schemes, several numerical examples – including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system – are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. This makes them suitable for adaptingto manifold-related computations that arise from high-dimensional molecular dynamics simulations.
Keywords: Symmetric decomposition; non-equilibrium thermodynamics; enhancement by mixture; exponential ergodicity; structure-preserving numerical scheme.