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

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.