Dynamique de Langevin

En physique, la dynamique de Langevin est une approche de la modélisation mathématique de la dynamique des systèmes moléculaires, développée à l'origine par le physicien français Paul Langevin. L'approche se caractérise par l'utilisation de modèles simplifiés basés sur des équations différentielles stochastiques. Aperçu Un système moléculaire dans le monde réel ne peut être présent dans le vide. Les molécules d'air causent des frottements et les collisions à grande vitesse qui se produisent perturbent le système. La dynamique de Langevin tente d'étendre la dynamique moléculaire pour prendre en compte ces effets. Elle permet aussi de contrôler la température comme avec un thermostat (ensemble canonique). La dynamique de Langevin reproduit l'aspect visqueux des solvants. Elle ne modélise pas complètement un solvant implicite, le modèle ne tenant pas compte de l'écrantage électrostatique ni de l'effet hydrophobe. Il est aussi important de noter que pour des solvants plus dens

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Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain

Elisabeth Agoritsas

As an extension of the isotropic setting presented in the companion paper Agoritsas et al (2019 J. Phys. A: Math. Theor. 52 144002), we consider the Langevin dynamics of a many-body system of pairwise interacting particles in d dimensions, submitted to an external shear strain. We show that the anisotropy introduced by the shear strain can be simply addressed by moving into the co-shearing frame, leading to simple dynamical mean field equations in the limit d -> infinity. The dynamics is then controlled by a single one-dimensional effective stochastic process which depends on three distinct strain-dependent kernels-self-consistently determined by the process itself-encoding the effective restoring force, friction and noise terms due to the particle interactions. From there one can compute dynamical observables such as particle mean-square displacements and shear stress fluctuations, and eventually aim at providing an exact d -> infinity benchmark for liquid and glass rheology. As an application of our results, we derive dynamically the 'statefollowing' equations that describe the static response of a glass to a finite shear strain until it yields.

2019

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Out-of-equilibrium dynamical equations of infinite-dimensional particle systems I. The isotropic case

Elisabeth Agoritsas

We consider the Langevin dynamics of a many-body system of interacting particles in d dimensions, in a very general setting suitable to model several out-of-equilibrium situations, such as liquid and glass rheology, active self-propelled particles, and glassy aging dynamics. The pair interaction potential is generic, and can be chosen to model colloids, atomic liquids, and granular materials. In the limit d -> infinity, we show that the dynamics can be exactly reduced to a single one-dimensional effective stochastic equation, with an effective thermal bath described by kernels that have to be determined self-consistently. We present two complementary derivations, via a dynamical cavity method and via a path-integral approach. From the effective stochastic equation, one can compute dynamical observables such as pressure, shear stress, particle mean-square displacement, and the associated response function. As an application of our results, we derive dynamically the 'state-following' equations that describe the response of a glass to quasistatic perturbations, thus bypassing the use of replicas. The article is written in a modular way, that allows the reader to skip the details of the derivations and focus on the physical setting and the main results.

2019

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Supersymmetries in nonequilibrium Langevin dynamics

Elisabeth Agoritsas

Stochastic phenomena are often described by Langevin equations, which serve as a mesoscopic model for microscopic dynamics. It has been known since the work of Parisi and Sourlas that reversible (or equilibrium) dynamics present supersymmetries (SUSYs). These are revealed when the path-integral action is written as a function not only of the physical fields, but also of Grassmann fields representing a Jacobian arising from the noise distribution. SUSYs leave the action invariant upon a transformation of the fields that mixes the physical and the Grassmann ones. We show that contrary to common belief, it is possible to extend the known reversible construction to the case of arbitrary irreversible dynamics, for overdamped Langevin equations with additive white noise—provided their steady state is known. The construction is based on the fact that the Grassmann representation of the functional determinant is not unique, and can be chosen so as to present a generalization of the Parisi-Sourlas SUSY. We show how such SUSYs are related to time-reversal symmetries and allow one to derive modified fluctuation-dissipation relations valid in nonequilibrium. We give as a concrete example the results for the Kardar-Parisi-Zhang equation.

2021

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Équation de Langevin

Léquation de Langevin' (1908) est une équation stochastique pour le mouvement brownien. : \underbrace{m\frac{d^2x(t)}{d^2t}}_{\text{force inertielle}}=\underbrace{-6 \pi \eta R \frac{

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