All- and one-particle distribution functions at nonequilibrium steady state under thermal gradient

: We provide a concrete expression for the phase-space distribution function at nonequilibrium steady state under a constant thermal gradient, which is a typical system of the nonequilibrium molecular dynamics simulation of heat conduction. First, the phase-space distribution function of all particles in a local volume is formulated. Our formulation explicitly takes into account the entropy production due to the change in equilibrium thermodynamic variables in addition to the traditional entropy production described by the spatial gradients and fluxes of equilibrium thermodynamic variables. This treatment is necessary to explain the nonequilibrium response of a quantity that has no equilibrium correlation with mass and heat fluxes and is essential to correctly deduce one-particle distribution functions from the all-particle one. From the all-particle distribution function, we derive the Green-Kubo relations that express the one-particle distribution functions of density and velocity in terms of equilibrium correlation functions and verify these expressions using the molecular dynamics simulation of a Lennard-Jones liquid. These nonequilibrium one-particle distribution functions are sufficiently tractable for practical use, such as for the analytical evaluation of the nonequilibrium average of physical quantities.