On the Numerical Stationary Distribution of Overdamped Langevin Equation in Harmonic System
arXiv:2211.14027 · 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 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.
arXiv comment19 pages
References in corpus (15)
- A simple and effective Verlet-type algorithm for simulating Langevin dynamics
- Robust and efficient configurational molecular sampling via Langevin Dynamics
- Rational Construction of Stochastic Numerical Methods for Molecular Sampling
- A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat
- The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics
- A unified thermostat scheme for efficient configurational sampling for classical/quantum canonical ensembles via molecular dynamics
- Complete Set of Stochastic Verlet-Type Thermostats for Correct Langevin Simulations
- Generating Transition Paths by Langevin Bridges
- Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics
- On the long-time integration of stochastic gradient systems
- Understanding Molecular Dynamics with Stochastic Processes via Real or Virtual Dynamics
- Postprocessed integrators for the high order integration of ergodic SDEs
- Time correlation functions of equilibrium and nonequilibrium Langevin dynamics: Derivations and numerics using random numbers
- Sampling constrained stochastic trajectories using Brownian bridges
- Analysis of Transition Path Ensemble in the Exactly Solvable Models via Overdamped Langevin Equation