Continuum and lattice heat currents for oscillator chains

TL;DR

This paper compares two definitions of heat current in oscillator chains, revealing differences in calculated heat conductivity that diminish with chain length, especially relevant for small structures.

Contribution

It demonstrates that two common heat current definitions yield different results for finite oscillator chains, highlighting the importance of choice in small systems.

Findings

Difference in heat conductivity ratios scales as 1/N for tethered chains

Difference decays as 1/N^eta for pressure-held chains, with 0.5 < eta < 1

Differences are significant for small structures and diminish as system size increases

Abstract

We show that two commonly used definitions for the heat current give different results--through the Kubo formula--for the heat conductivity of oscillator chains. The difference exists for finite chains, and is expected to be important more generally for small structures. For a chain of N particles that are tethered at the ends, the ratio of the heat conductivities calculated with the two currents differs from unity by O(1/N). For a chain held at constant pressure, the difference from unity decays more slowly, and is consistent with O(1/N^eta) with 1 > eta > 0.5.