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We study numerical methods for the generalized Langevin equation (GLE) with a positive Prony series memory kernel, in which case the GLE can be written in an extended variable Markovian formalism. We propose a new splitting method that is…

Computational Physics · Physics 2022-05-31 Manh Hong Duong , Xiaocheng Shang

We propose a generalized Langevin dynamics (GLD) technique to construct non-Markovian particle-based coarse-grained models from fine-grained reference simulations and to efficiently integrate them. The proposed GLD model has the form of a…

Soft Condensed Matter · Physics 2018-11-16 Gerhard Jung , Martin Hanke , Friederike Schmid

We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space…

Numerical Analysis · Mathematics 2020-12-09 Benedict Leimkuhler , Matthias Sachs

The Generalized Langevin Equation (GLE) has been recently suggested to simulate the time evolution of classical solid and molecular systems when considering general non-equilibrium processes. In this approach, a part of the whole system (an…

Statistical Mechanics · Physics 2014-04-23 L. Stella , C. D. Lorenz , L. Kantorovich

The complexity of molecular dynamics simulations necessitates dimension reduction and coarse-graining techniques to enable tractable computation. The generalized Langevin equation (GLE) describes coarse-grained dynamics in reduced…

Computational Physics · Physics 2020-06-08 Francesca Grogan , Huan Lei , Xiantao Li , Nathan A. Baker

Memory effects are ubiquitous in a wide variety of complex physical phenomena, ranging from glassy dynamics and metamaterials to climate models. The Generalised Langevin Equation (GLE) provides a rigorous way to describe memory effects via…

Disordered Systems and Neural Networks · Physics 2023-06-29 Max Kerr Winter , Ilian Pihlajamaa , Vincent E. Debets , Liesbeth M. C. Janssen

We present efficient finite difference estimators for goal-oriented sensitivity indices with applications to the generalized Langevin equation (GLE). In particular, we apply these estimators to analyze an extended variable formulation of…

Numerical Analysis · Mathematics 2018-09-18 Eric Joseph Hall , Markos A. Katsoulakis , Luc Rey-Bellet

Finding the dynamical law of observable quantities lies at the core of physics. Within the particular field of statistical mechanics, the generalized Langevin equation (GLE) comprises a general model for the evolution of observables…

Statistical Mechanics · Physics 2022-11-22 Antonio Russo , Miguel A. Duran-Olivencia , Ioannis G. Kevrekidis , Serafim Kalliadasis

The generalized Langevin equation is a model for the motion of coarse-grained particles where dissipative forces are represented by a memory term. The numerical realization of such a model requires the implementation of a stochastic…

Soft Condensed Matter · Physics 2021-05-26 Niklas Bockius , Jeanine Shea , Gerhard Jung , Friederike Schmid , Martin Hanke

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a Generalized Langevin Equation (GLE) for a general interacting many-body system. The resulting…

Chemical Physics · Physics 2022-08-31 Cihan Ayaz , Benjamin A. Dalton , Roland R. Netz

The generalized Langevin equation is used as a model for various coarse-grained physical processes, e.g., the time evolution of the velocity of a given larger particle in an implicitly represented solvent, when the relevant time scales of…

Statistical Mechanics · Physics 2025-11-13 Niklas Bockius , Maximilian Braun , Kay Hofmann , Friederike Schmid , Martin Hanke

We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of smooth observables in high-dimensional nonlinear systems with local interactions. Building…

Numerical Analysis · Mathematics 2020-03-18 Yuanran Zhu , Daniele Venturi

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the velocity of microparticles in viscoelastic fluids. In this work, we consider the large-time asymptotic properties of a…

Probability · Mathematics 2020-01-30 Nathan Glatt-Holtz , David Herzog , Scott McKinley , Hung Nguyen

Generalized Langevin Equation (GLE) thermostats have been used very effectively as a tool to manipulate and optimize the sampling of thermodynamic ensembles and the associated static properties. Here we show that a similar, exquisite level…

Chemical Physics · Physics 2017-09-06 Mariana Rossi , Venkat Kapil , Michele Ceriotti

Fundamental understanding of complex dynamics in many-particle systems on the atomistic level is of utmost importance. Often the systems of interest are of macroscopic size but can be partitioned into few important degrees of freedom which…

Chemical Physics · Physics 2015-07-09 Fabian Gottwald , Sven Karsten , Sergei D. Ivanov , Oliver Kühn

The Generalized Langevin Equation (GLE) is a Stochastic Integro-Differential Equation that is commonly used to describe the velocity of microparticles that move randomly in viscoelastic fluids. Such particles commonly exhibit what is known…

Probability · Mathematics 2017-11-03 Scott A McKinley , Hung D Nguyen

We introduce a novel approach for learning memory kernels in Generalized Langevin Equations. This approach initially utilizes a regularized Prony method to estimate correlation functions from trajectory data, followed by regression over a…

Machine Learning · Statistics 2025-05-22 Quanjun Lang , Jianfeng Lu

In this paper, we study a non-Markovian generalized relativistic Langevin equation (GRLE). We show that when the memory kernel is a sum of exponentials, the GRLE is equivalent to a Markovian system with added variables. We establish the…

Probability · Mathematics 2026-03-17 Ethan Baker , Manh Hong Duong , Hung Dang Nguyen

We analyze prediction error in stochastic dynamical systems with memory, focusing on generalized Langevin equations (GLEs) formulated as stochastic Volterra equations. We establish that, under a strongly convex potential, trajectory…

Machine Learning · Statistics 2025-12-12 Quanjun Lang , Jianfeng Lu

We study the statistically invariant structures of the nonlinear generalized Langevin equation (GLE) with a power-law memory kernel. For a broad class of memory kernels, including those in the subdiffusive regime, we construct solutions of…

Probability · Mathematics 2023-01-10 David P. Herzog , Jonathan C. Mattingly , Hung D. Nguyen
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