Theory of applying heat flow from thermostatted boundary walls: Dissipative and local-equilibrium responses and fluctuation theorems

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 B in terms of the time-correlation functions between B and the heat flows JK from the thermostats to the particles. Furthermore, the surface variables JK 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.