Linear response formula for finite frequency thermal conductance of open systems

TL;DR

This paper derives an exact linear response formula for the thermal conductance of open systems at finite frequencies, providing insights into frequency-dependent heat transport in oscillator models.

Contribution

It presents a novel exact linear response expression valid at all frequencies and applies it to analyze frequency-dependent thermal conductance in oscillator chains.

Findings

Momentum conserving systems show a low frequency peak in response.

Momentum non-conserving systems lack a low frequency peak.

The analytical response matches numerical simulations very well.

Abstract

An exact linear response expression is obtained for the heat current in a classical Hamiltonian system coupled to heat baths with time-dependent temperatures. The expression is equally valid at zero and finite frequencies. We present numerical results on the frequency dependence of the response function for three different one-dimensional models of coupled oscillators connected to Langevin baths with oscillating temperatures. For momentum conserving systems, a low frequency peak is seen that, is higher than the zero frequency response for large systems. For momentum non-conserving systems, there is no low frequency peak. The momentum non-conserving system is expected to satisfy Fourier's law, however, at the single bond level, we do not see any clear agreement with the predictions of the diffusion equation even at low frequencies. We also derive an exact analytical expression for the response of a chain of harmonic oscillators to a (not necessarily small) temperature difference; the agreement with the linear response simulation results for the same system is excellent.