On numerical stationary distribution of overdamped Langevin equation in harmonic system

Abstract

Efficient numerical algorithm for stochastic differential equation has been an important object in the research of statistical physics and mathematics for a long time. In this work we study the highly accurate numerical algorithm for the overdamped Langevin equation. In particular, our interest is in the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system. Based on the large friction limit of the underdamped Langevin dynamic scheme, three algorithms for overdamped Langevin equation are obtained. We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory for both one-dimensional case and multi-dimensional case. The accuracy of the stationary distribution of each algorithm is illustrated by comparing with the exact Boltzmann distribution. Our results demonstrate that the “BAOA-limit” algorithm generates an accurate distribution of the harmonic system in a canonical ensemble, within a stable range of time interval. The other algorithms do not produce the exact distribution of the harmonic system.