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

We present a generalization of the Green-Kubo expressions for thermal transport coefficients μ in complex fluids of the generic form μ=μ_∞+∫₀^∞dtV⁻¹⟨J_ϵ exp(tL)J⟩₀, i.e., a sum of an instantaneous transport coefficient μ_∞, and a time integral over a time correlation function in a state of thermal equilibrium between a current J and its conjugate current J_ϵ. The streaming operator exp(tL) generates the trajectory of a dynamical variable J(t)=exp(tL)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_∞≠0 and J_ϵ≠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.