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The transport of individual particles in inhomogeneous environments is complex and exhibits non-Markovian responses. The latter may be quantified by a memory function within the framework of the linear generalised Langevin equation (GLE).…

Soft Condensed Matter · Physics 2024-07-08 Arthur V. Straube , Felix Höfling

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

We present a statistical mechanics framework for modeling equilibrium friction coefficients using the Generalized Langevin Equation (GLE). We show that the kernel, obtained via the Fluctuation-Dissipation Theorem (FDT) from the stochastic…

Plasma Physics · Physics 2026-05-01 N. R. Sree Harsha , Zhenyuan Yu , Chuang Ren , Virginia Billings , Michael Huang

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 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, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly…

Numerical Analysis · Mathematics 2022-09-20 Di Fang , Lei Li

Local diffusivity of a protein depends crucially on the conformation, and the conformational fluctuations are often non-Markovian. Here, we investigate the Langevin equation with non-Markovian fluctuating diffusivity, where the fluctuating…

Statistical Mechanics · Physics 2023-04-26 Mutsumi Kimura , Takuma Akimoto

This work puts forward a generalization of the well-known rocking Markovian Brownian ratchets to the realm of antipersistent non-Markovian subdiffusion in viscoelastic media. A periodically forced subdiffusion in a parity-broken ratchet…

Statistical Mechanics · Physics 2014-09-24 Igor Goychuk

Single-file diffusion behaves as normal diffusion at small time and as anomalous subdiffusion at large time. These properties can be described by fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We…

Statistical Mechanics · Physics 2015-05-13 S. C. Lim , L. P. Teo

Motivated by subdiffusive motion of bio-molecules observed in living cells we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and…

Statistical Mechanics · Physics 2016-09-08 Jae-Hyung Jeon , Ralf Metzler

We consider the fractional Langevin equation far from equilibrium (FLEFE) to describe stochastic dynamics which do not obey the fluctuation-dissipation theorem, unlike the conventional fractional Langevin equation (FLE). The solution of…

Statistical Mechanics · Physics 2024-12-17 Qing Wei , Wei Wang , Yifa Tang , Ralf Metzler , Aleksei Chechkin

Brownian yet non-Gaussian phenomenon has recently been observed in many biological and active matter systems. The main idea of explaining this phenomenon is to introduce a random diffusivity for particles moving in inhomogeneous…

Statistical Mechanics · Physics 2022-01-19 Xudong Wang , Yao Chen

The Markovian diffusion theory is generalized within the framework of the special theory of relativity using a modification of the mathematical calculus of diffusion on Riemannian manifolds (with definite metric) to describe diffusion on…

Mathematical Physics · Physics 2013-05-29 Joachim Herrmann

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though origins of these behaviors are still elusive. Here, as a model to describe such…

Statistical Mechanics · Physics 2016-07-13 Tomoshige Miyaguchi , Takuma Akimoto , Eiji Yamamoto

For reproducing the anomalous -- i.e., sub- or super-diffusive -- behavior in some stochastic dynamical systems, the Generalized Langevin Equation (GLE) has gained considerable popularity in recent years. Motivated by the question whether…

Soft Condensed Matter · Physics 2010-02-05 Debabrata Panja

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 (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate…

Numerical Analysis · Mathematics 2026-04-29 Xinjie Dai , Xingyu Liu , Diancong Jin , Liying Sun

Recent rapid advances in single particle tracking and supercomputing techniques resulted in an unprecedented abundance of diffusion data exhibiting complex behaviours, such the presence of power law tails of the msd and memory functions,…

Statistical Mechanics · Physics 2018-10-08 Jakub Ślęzak

We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency.…

Statistical Mechanics · Physics 2015-07-20 T. Srokowski , A. Kaminska

We present a comparative study on two theoretical descriptions of microrheological experiments. Using a generalized Langevin equation (GLE), we analyze the origin of the power-law behavior of the main properties of a viscoelastic medium.…

Statistical Mechanics · Physics 2009-11-13 I. Santamaria-Holek