# Archive ouverte HAL – Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain

### Elisabeth Agoritsas 1 Thibaud Maimbourg 2 Francesco Zamponi 3

#### Elisabeth Agoritsas, Thibaud Maimbourg, Francesco Zamponi. Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2019, 52 (33), pp.334001. ⟨10.1088/1751-8121/ab2b68⟩. ⟨hal-02291782⟩

As an extension of the isotropic setting presented in the companion paper [J. Phys. A 52, 144002 (2019)], 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 ${d\to\infty}$. 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 ${d \to \infty}$ 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.

• 1. EPFL - Ecole Polytechnique Fédérale de Lausanne
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
• 3. LPENS - UMR 8023 - Laboratoire de physique de l'ENS - ENS Paris