On numerical stationary distribution of overdamped Langevin equation in harmonic system

doi:10.1088/1674-1056/acc05c

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.

Keywords: numerical stationary distribution overdamped Langevin equation exact solution harmonic system

Received: 12 December 2022
Revised: 22 January 2023
Accepted manuscript online: 02 March 2023

PACS:	05.20.-y	(Classical statistical mechanics)
	05.10.Gg	(Stochastic analysis methods)
	02.30.Jr	(Partial differential equations)

Fund: Project supported by the Basic and Applied Basic Research Foundation of Guangdong Province, China (Grant No.2021A1515010328), the Key-Area Research and Development Program of Guangdong Province, China (Grant No.2020B010183001), and the National Natural Science Foundation of China (Grant No.12074126).

Corresponding Authors: Xiao-Bao Yang
E-mail: scxbyang@scut.edu.cn

Cite this article:

De-Zhang Li(李德彰) and Xiao-Bao Yang(杨小宝) On numerical stationary distribution of overdamped Langevin equation in harmonic system 2023 Chin. Phys. B 32 080501

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