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

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 à 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.

References in corpus (9)

• Unified study of glass and jamming rheology in soft particle systems
• Work and heat probability distribution of an optically driven Brownian particle: Theory and experiments
• Extended Fluctuation-Dissipation Theorem for Soft Matter in Stationary Flow
• Interacting Brownian dynamics in a nonequilibrium particle bath
• On the quantum description of Einstein's Brownian motion
• Radial distribution function of Lennard-Jones fluids in shear flows from intermediate asymptotics
• Generalized Langevin dynamics simulation with non-stationary memory kernels: How to make noise
• Microscopic theory for the pair correlation function of liquidlike colloidal suspensions under shear flow
• Solution to the two-body Smoluchowski equation with shear flow for charge-stabilized colloids at low to moderate Péclet numbers

Cited by in corpus (1)

• Exact results for non-Newtonian transport properties in sheared granular suspensions: inelastic Maxwell models and BGK-type kinetic model