Generalized Green-Kubo formulas for fluids with impulsive, dissipative, stochastic and conservative interactions

TL;DR

This paper generalizes Green-Kubo formulas to include complex fluids with various interaction types, providing a unified framework for calculating thermal transport coefficients in systems with conservative, impulsive, stochastic, and dissipative forces.

Contribution

It introduces a generalized Green-Kubo formula applicable to a wide range of fluid models, including non-conservative and stochastic interactions, extending the traditional approach.

Findings

Derived a unified Green-Kubo formula for complex fluids.

Applied the formula specifically to hard sphere fluids.

Showed that the instantaneous transport coefficient can be non-zero in general.

Abstract

We present a generalization of the Green-Kubo expressions for thermal transport coefficients $\mu$ in complex fluids of the generic form, $\mu= \mu_\infty +\int^\infty_0 dt V^{-1} < J_\epsilon \exp(t {\cal L}) J >_0$, i.e. a sum of an instantaneous transport coefficient $\mu_\infty$, and a time integral over a time correlation function in a state of thermal equilibrium between a current $J$ and a transformed current $J_\epsilon$. The streaming operator $\exp(t{\cal L})$ generates the trajectory of a dynamical variable $J(t) =\exp(t{\cal L}) J$ when used inside the thermal average $<...>_0$. These formulas are valid for conservative, impulsive (hard spheres), stochastic and dissipative forces (Langevin fluids), provided the system approaches a thermal equilibrium state. In general $\mu_\infty \neq 0$ and $J_\epsilon \neq J$, except for the case of conservative forces, where the equality signs apply. The most important application in the present paper is the hard sphere fluid.