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

TL;DR

This paper develops a microscopic theory for heat flow from thermostatted walls in fluid films, linking response functions to correlation functions, deriving steady-state distributions, and presenting fluctuation theorems.

Contribution

It introduces a general microscopic framework for heat flow response, connecting surface heat fluxes to bulk properties, and derives steady-state distributions and fluctuation theorems from first principles.

Findings

Response functions expressed via time-correlation functions.

Fourier's law derived with Green's thermal conductivity.

Steady-state distribution obtained in McLennan-Zubarev form.

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.