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

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2019-04, Vol.52 (14), p.144002
• Main Authors: Agoritsas, Elisabeth, Maimbourg, Thibaud, Zamponi, Francesco
• Format: Article
• Language: English
• Subjects: Condensed Matter; disordered systems; Disordered Systems and Neural Networks; mean field; metastable glassy states; out-of-equilibrium dynamics; Physics; rheology; Statistical Mechanics
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ab099d

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 , 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.