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

Abstract

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 → ∞, 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.

Brittle yielding of amorphous solids at finite shear rates

“…It would be interesting to understand and characterise better the lengthscale ξ in various theoretical settings, from atomistic simulations in various glassy models to more coarse-grained descriptions such as elastoplastic models where larger system sizes can more easily be studied, in particular perhaps in 3D. More generally, our work should motivate theoretical models, such as soft glassy rheology [63], shear transformation zone [68], elasto-plastic models [31], mode-coupling theory [71] and random first order transition theory [72,73] to address the problem of brittle yielding at finite shear rates.…”

Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids

“…In this work, we analyzed the DMFT equations for infinite-dimensional equilibrium liquids derived in [15][16][17]. We derived the DMFT equations for Hard Spheres as a limit of those for regular potentials, and we presented some methods to solve the DMFT equations numerically.…”

“…Despite its non-systematic nature [4][5][6][7], MCT accurately describes the initial slowing down of liquid dynamics upon supercooling, including the wavevector dependence of correlation functions [8].Interestingly, in the formal limit in which the spatial dimension d goes to infinity, liquid thermodynamics reduces to the calculation of the second virial coefficient [9][10][11][12]. Based on this observation, a first attempt to solve exactly liquid dynamics for d → ∞ was presented in [13] (see also [14]), but the full dynamical mean field theory (DMFT) that describes exactly the equilibrium dynamics in d → ∞ was only derived recently, via a second-order virial expansion on trajectories [15] or via a dynamic cavity method [16,17]. The DMFT provides a set of closed one-dimensional integro-differential equations, which exactly describe the many-body liquid dynamics in the thermodynamic limit, and are similar in structure to those obtained for quantum systems in the same limit [18].…”

“…The DMFT provides a set of closed one-dimensional integro-differential equations, which exactly describe the many-body liquid dynamics in the thermodynamic limit, and are similar in structure to those obtained for quantum systems in the same limit [18]. We refer the reader to [17,[19][20][21] for a detailed review of the solution of liquid dynamics in infinite dimensions, including its extension to the out-of-equilibrium setting.Unfortunately, the analytical solution of the DMFT equations is out of reach. In this work, we present their numerical solution, obtained through an iterative method and a straightforward discretization of time.…”

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

“…At last λ i (t) is an external field set to 0 except when we need to define the response function. We provide an extended discussion in the companion paper [1] about the different physical situations encompassed by this general setting, depending on the specific choices of the friction and noise kernels. Here we want to focus on the role of the fluid velocity v f (x, t), set to zero in Ref.…”