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

Abstract

As an extension of the isotropic setting presented in the companion paper [1], 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 → ∞. The dynamics is then controlled by a single one-dimensional effective stochastic process which depends on three distinct straindependent 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 → ∞ 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.

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

“…+ c − (s) = c + (s) , c − (s)λ − (s) = c + (s)λ + (s) , (B3) which imply c − (s) = λ + (s) s[λ − (s) − λ + (s)] , c + (s) = λ − (s) s[λ − (s) − λ + (s)] ,(B4)thus completing the proof of Eq (20)…”

“…This work opens the way to the numerical solution of the DMFT equations that describe non-equilibrium liquids [17,20], which hopefully will give insight on a variety of phenomena such as yielding, jamming, and glass melting, both in passive and active systems.…”

“…(19), the solution is given by Eq. (20) with yet unknown functions c ± (s). Now we should impose the continuity conditions in h 0 = 0.…”

“…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.…”

A direct link between active matter and sheared granular systems

“…These results qualitatively explain many features in sheared two- and three-dimensional glassy solids. Perhaps more interestingly, new work suggests that the dynamical mean-field equations in infinite dimensions have the same structure regardless of whether the driving forces are generated by global shear or active forces on each particle ( 6 , 15 – 17 ), as all such forcing can be represented by memory kernels with the same functional form.…”