Heat conductance in nonlinear lattices at small temperature gradients

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.