Nonstationary heat conduction in one-dimensional models with substrate potential

TL;DR

This study examines non-stationary heat conduction in one-dimensional models with substrate potential, revealing a crossover from oscillatory to diffusive decay, challenging classical Fourier-based models and highlighting complex relaxation behaviors.

Contribution

The paper demonstrates that hyperbolic models are necessary to describe heat conduction in these systems and shows that existing simple hyperbolic equations like the Cattaneo-Vernotte model are insufficient.

Findings

Crossover from oscillatory to diffusive decay observed in all models.

Relaxation of thermal perturbations is exponential, not power-law.

Behavior does not conform to traditional universality classes for heat conduction.

Abstract

The paper investigates non-stationary heat conduction in one-dimensional models with substrate potential. In order to establish universal characteristic properties of the process, we explore three different models --- Frenkel-Kontorova (FK), phi4+ ($\phi^4$+) and phi4- ($\phi^4-$). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with parabolic Fourier equation of the heat conduction and clearly demonstrates the necessity of hyperbolic models. The crossover wavelength decreases with increase of average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for temperature field is justified. In all model investigated, the relaxation of thermal perturbations is exponential -- contrary to linear chain, where it follows a power law. However, the most popular lowest-order hyperbolic generalization of the Fourier law, known as Cattaneo-Vernotte (CV) or telegraph equation (TE) is not valid for description of the observed behavior of the models with on-site potential. In part of the models a characteristic relaxation times exhibit peculiar scaling with respect to the temperature perturbation wavelength. Quite surprisingly, such behavior is similar to that of well-known model with divergent heat conduction (Fermi-Pasta-Ulam chain) and rather different from the model with normal heat conduction (chain of rotators). Thus, the data on the non-stationary heat conduction in the systems with on-site potential do not fit commonly accepted concept of universality classes for heat conduction in one-dimensional models.