Dissipation and Entropy Production in Deterministic Heat Conduction of Quasi-one-dimensional Systems

TL;DR

This paper investigates heat conduction in deterministic quasi-one-dimensional systems, deriving local heat flux, analyzing entropy production, and exploring how these properties scale with system size and temperature gradients.

Contribution

It introduces a microscopic framework for heat flux and entropy production in deterministic systems, revealing detailed mechanisms and scaling behaviors in nonequilibrium steady states.

Findings

Local entropy density closely matches from velocity distributions.

Entropy production near walls proportional to phase space contraction for equal reservoirs.

Scaling relations identified for local properties and boundary effects.

Abstract

We explore the consequences of a deterministic microscopic thermostat-reservoir contact mechanism. With different temperature reservoirs at each end of a two-dimensional system, a heat current is produced and the system has an anomalous thermal conductivity. The microscopic form for the local heat flux vector is derived and both the kinetic and potential contributions are calculated. The total heat flux vector is shown to satisfy the continuity equation. The properties of this nonequilibrium steady state are studied as a function of system size and temperature gradient identifying key scaling relations for the local fluid properties and separating bulk and boundary effects. The local entropy density calculated from the local equilibrium distribution is shown to be a very good approximation to the entropy density calculated directly from velocity distribution even for systems that are far from equilibrium. The dissipation and kinetic entropy production and flux are compared quantitatively and the differing mechanisms discussed within the BGY approximation. For equal temperature reservoirs the entropy production near the reservoir walls is shown to be proportional to the local phase space contraction calculated from the tangent space dynamics. However, for unequal temperatures, the connection between local entropy production and local phase space contraction is more complicated.