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A wide variety of numerical methods are evaluated and compared for solving the stochastic differential equations encountered in molecular dynamics. The methods are based on the application of deterministic impulses, drifts, and Brownian…

Computational Physics · Physics 2013-05-14 Benedict Leimkuhler , Charles Matthews

We consider numerical methods for thermodynamic sampling, i.e. computing sequences of points distributed according to the Gibbs-Boltzmann distribution, using Langevin dynamics and overdamped Langevin dynamics (Brownian dynamics). A wide…

Statistical Mechanics · Physics 2015-01-13 Benedict Leimkuhler , Charles Matthews , Gabriel Stoltz

We introduce a numerical method for Brownian dynamics with position dependent diffusion tensor which is second order accurate for sampling the invariant measure while requiring only one force evaluation per timestep. Analysis of the…

Numerical Analysis · Mathematics 2025-01-09 Eugen Bronasco , Benedict Leimkuhler , Dominic Phillips , Gilles Vilmart

Adaptive Langevin dynamics is a method for sampling the Boltzmann-Gibbs distribution at prescribed temperature in cases where the potential gradient is subject to stochastic perturbation of unknown magnitude. The method replaces the…

Probability · Mathematics 2023-11-14 Benedict Leimkuhler , Matthias Sachs , Gabriel Stoltz

We study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. Extended stochastic dynamical methods are discussed which have application to…

Numerical Analysis · Mathematics 2016-03-08 Benedict Leimkuhler , Xiaocheng Shang

We present a new method of conducting molecular dynamics simulation in isothermal-isobaric ensemble based on Langevin equations of motion. The stochastic coupling to all particle and cell degrees of freedoms is introduced in a correct way,…

Statistical Mechanics · Physics 2016-04-28 Xingyu Gao , Jun Fang , Han Wang

We propose a sampling method based on an ensemble approximation of second order Langevin dynamics. The log target density is appended with a quadratic term in an auxiliary momentum variable and damped-driven Hamiltonian dynamics introduced;…

Dynamical Systems · Mathematics 2025-06-06 Ziming Liu , Andrew M. Stuart , Yixuan Wang

We present a novel method for drawing samples from Gibbs distributions with densities of the form $\pi(x) \propto \exp(-U(x))$. The method accelerates the unadjusted Langevin algorithm by introducing an inertia term similar to Polyak's…

Numerical Analysis · Mathematics 2025-10-09 Alexander Falk , Andreas Habring , Christoph Griesbacher , Thomas Pock

We show a unified second-order scheme for constructing simple, robust and accurate algorithms for typical thermostats for configurational sampling for the canonical ensemble. When Langevin dynamics is used, the scheme leads to the BAOAB…

Chemical Physics · Physics 2017-10-17 Zhijun Zhang , Xinzijian Liu , Zifei Chen , Haifeng Zheng , Kangyu Yan , Jian Liu

We present a method for sampling microscopic configurations of a physical system distributed according to a canonical (Boltzmann-Gibbs) measure, with a constraint holding in average. Assuming that the constraint can be controlled by the…

Statistical Mechanics · Physics 2008-10-03 Jean-Bernard Maillet , Gabriel Stoltz

The molecular dynamics lattice gas method maps a molecular dynamics simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method by taking a Boltzmann average…

Computational Physics · Physics 2021-10-04 Aleksandra Pachalieva , Alexander J. Wagner

We present a method for computing parameter sensitivities and response coefficients in Brownian dynamics simulations. The method involves tracking auxiliary variables (Malliavin weights) in addition to the usual particle positions, in an…

Statistical Mechanics · Physics 2012-07-20 Patrick B. Warren , Rosalind J. Allen

This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic…

Numerical Analysis · Mathematics 2015-06-18 B. Leimkuhler , C. Matthews , M. V. Tretyakov

Two recent landmark experiments have performed Gaussian boson sampling (GBS) with a non-programmable linear interferometer and threshold detectors on up to 144 output modes (see Refs.~\onlinecite{zhong_quantum_2020,zhong2021phase}). Here we…

In previous work, we introduced a method for determining convergence rates for integration methods for the kinetic Langevin equation for $M$-$\nabla$Lipschitz $m$-log-concave densities [arXiv:2302.10684, 2023]. In this article, we exploit…

Numerical Analysis · Mathematics 2023-06-16 Benedict Leimkuhler , Daniel Paulin , Peter A. Whalley

Two of the most widely used Langevin integrators for molecular dynamics simulations are the GROMACS Stochastic Dynamics (GSD) integrator and the splitting method BAOAB. In this letter, we show that the GROMACS Stochastic Dynamics integrator…

Computational Physics · Physics 2022-09-28 Stefanie Kieninger , Bettina G. Keller

Adsorption is a relevant process in many fields, such as product manufacturing or pollution remediation in porous materials. Adsorption takes place at the molecular scale, amenable to be modeled by Lagrangian numerical methods. We have…

Machine Learning · Statistics 2020-04-15 Maryam Rahbaralam , Amir Abdollahi , Daniel Fernàndez-Garcia , Xavier Sanchez-Vila

Langevin dynamics has become a popular tool to simulate the Boltzmann equilibrium distribution. When the repartition of the Langevin equation involves the exact realization of the Ornstein-Uhlenbeck noise, in addition to the conventional…

Chemical Physics · Physics 2017-11-15 Dezhang Li , Xu Han , Yichen Chai , Cong Wang , Zifei Chen , Zhijun Zhang , Jian Liu , Jiushu Shao

We derive and analyze numerical methods for underdamped (kinetic) Langevin dynamics in a domain with elastic reflection at the boundary. First-order approximations are based on an Euler-type scheme incorporating collision-handling at the…

Numerical Analysis · Mathematics 2025-12-10 B. Leimkuhler , A. Sharma , M. V. Tretyakov

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the…

Numerical Analysis · Mathematics 2024-04-22 Dominic Phillips , Charles Matthews , Benedict Leimkuhler
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