Heat conductance in nonlinear lattices at small temperature gradients

TL;DR

This paper introduces a new framework for studying heat conductance in 1D nonlinear lattices, revealing a temperature-dependent transition between phonon and soliton mechanisms, and providing a computationally efficient approach for small temperature gradients.

Contribution

A novel methodological framework separates equilibrium and non-equilibrium heat conductance, enabling efficient analysis and identifying a temperature threshold for different heat transfer mechanisms.

Findings

Threshold temperature scales as N^{-3} with lattice size.

Phonon mechanism dominates below T_thr, solitons above.

Visualization of solitons and breathers in numerical experiments.

Abstract

This paper proposes a new methodological framework within which the heat conductance in 1D lattices can be studied. The total process of heat conductance is separated into two parts where the first one is the equilibrium process at equal temperatures $T$ of both ends and the second one -- non-equilibrium with the temperature $\Delta T$ of one end and zero temperature of the other. This approach allows significant decrease of computational time at $\Delta T \to 0$. The threshold temperature $T_{\rm thr}$ is found which scales $T_{\rm thr}(N) \sim N^{-3}$ with the lattice size $N$ and by convention separates two mechanisms of heat conductance: phonon mechanism dominates at $T < T_{\rm thr}$ and the soliton contribution increases with temperature at $T > T_{\rm thr}$. Solitons and breathers are directly visualized in numerical experiments. The problem of heat conductance in non-linear lattices in the limit $\Delta T \to 0$ can be reduced to the heat conductance of harmonic lattice with time-dependent stochastic rigidities determined by the equilibrium process at temperature $T$. The detailed analysis is done for the $\beta$-FPU lattice though main results are valid for one-dimensional lattices with arbitrary potentials.