On the Numerical Stationary Distribution of Overdamped Langevin Equation in Harmonic System

TL;DR

This paper investigates the stationary distributions of numerical algorithms solving the overdamped Langevin equation in harmonic systems, revealing that the BAOA-limit algorithm exactly reproduces the Boltzmann distribution within stability limits.

Contribution

It derives explicit stationary distributions for three algorithms and identifies that the BAOA-limit algorithm uniquely yields the exact distribution for harmonic systems.

Findings

BAOA-limit algorithm produces the exact stationary distribution in harmonic systems.

Other algorithms do not generate the exact distribution.

The analysis covers both one-dimensional and multi-dimensional cases.

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 paper we study the highly accurate numerical algorithm of the overdamped Langevin equation. In particular, our interest is the behaviour of the numerical schemes for solving the overdamped Langevin equation in the harmonic system. Three algorithms are obtained for overdamped Langevin equation, from the large friction limit of the schemes for underdamped Langevin dynamics. We derive the explicit expression of the stationary distribution of each algorithm by analysing the discrete time trajectory, for both one-dimensional and multi-dimensional cases. The accuracy of the stationary distribution of each algorithm is illustrated by comparing to the exact Boltzmann distribution. Our results demonstrate that, the "BAOA-limit" algorithm generates the exact distribution for the harmonic system in the canonical ensemble, within the stable regime of the time interval. The other algorithms do not produce the exact distribution of the harmonic system.