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

Abstract

We present a generalization of the Green-Kubo expressions for thermal transport coefficients $μ$ in complex fluids of the generic form, $μ= μ_\infty +\int^\infty_0 dt V^{-1} < J_ε\exp(t {\cal L}) J >_0$, i.e. a sum of an instantaneous transport coefficient $μ_\infty$, and a time integral over a time correlation function in a state of thermal equilibrium between a current $J$ and a transformed current $J_ε$. 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 $μ_\infty \neq 0$ and $J_ε\neq J$, 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.