On numerical stationary distribution of overdamped Langevin equation in harmonic system

doi:10.1088/1674-1056/acc05c

中国物理B ›› 2023, Vol. 32 ›› Issue (8): 80501-080501.doi: 10.1088/1674-1056/acc05c

On numerical stationary distribution of overdamped Langevin equation in harmonic system

De-Zhang Li(李德彰) and Xiao-Bao Yang(杨小宝)^†

Department of Physics, South China University of Technology, Guangzhou 510640, China

收稿日期:2022-12-12 修回日期:2023-01-22 接受日期:2023-03-02 发布日期:2023-07-14
通讯作者: Xiao-Bao Yang E-mail:scxbyang@scut.edu.cn
基金资助:
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).

On numerical stationary distribution of overdamped Langevin equation in harmonic system

De-Zhang Li(李德彰) and Xiao-Bao Yang(杨小宝)^†

Department of Physics, South China University of Technology, Guangzhou 510640, China

Received:2022-12-12 Revised:2023-01-22 Accepted:2023-03-02 Published:2023-07-14
Contact: Xiao-Bao Yang E-mail:scxbyang@scut.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: numerical stationary distribution, overdamped Langevin equation, exact solution, 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.

Key words: numerical stationary distribution, overdamped Langevin equation, exact solution, harmonic system

中图分类号: (Classical statistical mechanics)

05.20.-y

05.10.Gg (Stochastic analysis methods) 02.30.Jr (Partial differential equations)

De-Zhang Li(李德彰) and Xiao-Bao Yang(杨小宝). On numerical stationary distribution of overdamped Langevin equation in harmonic system[J]. 中国物理B, 2023, 32(8): 80501-080501.

De-Zhang Li(李德彰) and Xiao-Bao Yang(杨小宝). On numerical stationary distribution of overdamped Langevin equation in harmonic system[J]. Chin. Phys. B, 2023, 32(8): 80501-080501.

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On numerical stationary distribution of overdamped Langevin equation in harmonic system

On numerical stationary distribution of overdamped Langevin equation in harmonic system

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