Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian

TL;DR

This paper derives a first-principles generalized Langevin equation for particles under shear flow, revealing its non-Markovian nature at higher shear rates and establishing the conditions under which simplified models are valid.

Contribution

It provides a first-principles derivation of the Langevin equation with shear flow from a Caldeira-Leggett Hamiltonian, clarifying the validity of common approximations.

Findings

Standard Langevin equation is valid only at very weak shear rates.

At higher shear rates, the Langevin dynamics become strongly non-Markovian.

Derived fluctuation-dissipation theorems applicable to sheared systems.

Abstract

We provide a first-principles derivation of the Langevin equation with shear flow and its corresponding fluctuation-dissipation theorems. Shear flow of simple fluids has been widely investigated by numerical simulations. Most studies postulate a Markovian Langevin equation with a simple shear drag term \`a la Stokes. However, this choice has never been justified from first principles. We start from a particle-bath system described by a classical Caldeira-Leggett Hamiltonian modified by adding a term proportional to the strain-rate tensor according to Hoover's DOLLS method, and we derive a generalized Langevin equation for the sheared system. We then compute, analytically, the noise time-correlation functions in different regimes. Based on the intensity of the shear-rate, we can distinguish between close-to-equilibrium and far-from-equilibrium states. According to the results presented here, the standard, simple and Markovian form of the Langevin equation with shear flow postulated in the literature is valid only in the limit of extremely weak shear rates compared to the effective vibrational temperature of the bath. For even marginally higher shear rates, the (generalized) Langevin equation is strongly non-Markovian and non-trivial fluctuation-dissipation theorems are derived.