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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 . 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 benchmark for liquid and glass rheology. As an application of our results, we derive dynamically the ‘state-following’ equations that describe the static response of a glass to a finite shear strain until it yields. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Systems of particles interacting via inverse-power law potentials have an invariance with respect to changes in length and temperature, implying a correspondence in the dynamics and thermodynamics between different “isomorphic” sets of temperatures and densities. In a recent series of works, it has been argued that such correspondences hold to a surprisingly good approximation in a much more general class of potentials, an observation that summarizes many properties that have been observed in the past. In this paper we show that such relations are exact in high-dimensional liquids and glasses, a limit in which the conditions for these mappings to hold become transparent. The special role played by the exponential potential is also confirmed. [ABSTRACT FROM AUTHOR]