Temperature relaxation and the Kapitza boundary resistance paradox

TL;DR

This paper resolves the longstanding paradox in calculating Kapitza boundary resistance between similar solids by deriving an exact formula based on temperature differences in the final state, clarifying the analogy to ballistic electron transport.

Contribution

It introduces a novel approach to calculating boundary resistance by focusing on the final heat-transporting state, eliminating the paradox present in previous methods.

Findings

Derived an exact, paradox-free formula for boundary resistance in a 1D model

Clarified the analogy between phonon boundary resistance and ballistic electron transport

Resolved the paradox of finite resistance in identical solids

Abstract

The calculation of the Kapitza boundary resistance between dissimilar harmonic solids has since long (Little [Can. J. Phys. 37, 334 (1959)]) suffered from a paradox: this resistance erroneously tends to a finite value in the limit of identical solids. We resolve this paradox by calculating temperature differences in the final heat-transporting state, rather than with respect to the initial state of local equilibrium. For a one-dimensional model we thus derive an exact, paradox-free formula for the boundary resistance. The analogy to ballistic electron transport is explained.