Theory of Applying Heat Flow from Thermostatted Boundary Walls: Dissipative and Local-Equilibrium Responses and Fluctuation Theorems

Abstract

We construct a microscopic theory of applying a heat flow from thermostatted boundary walls in the film geometry. We treat a classical one-component fluid, but our method is applicable to any fluids and solids. We express linear response of any variable ${\cal B}$ in terms of the time-correlation functions between $\cal B$ and the heat flows ${\cal J}_K$ from the thermostats to the particles. Furthermore, the surface variables ${\cal J}_K$ can be written in the form of space integrals of bulk quantities from the equations of motion. Owing to this surface-to-bulk relation, the steady-state response functions consist of dissipative and local-equilibrium parts, where the former gives rise to Fourier's law with Green's expression for the thermal conductivity. In the nonlinear regime, we derive the steady-state distribution in the phase space in the McLennan-Zubarev form from the first principles. Some fluctuation theorems are also presented.