High-frequency thermal processes in harmonic crystals

TL;DR

This paper analytically investigates high-frequency thermal processes in harmonic crystals, revealing that temperature anisotropy persists at equilibrium and providing formulas for energy redistribution and equilibration timescales.

Contribution

It derives equations describing high-frequency thermal processes in harmonic crystals and analytically characterizes their relaxation times and stationary temperature tensor anisotropy.

Findings

Characteristic relaxation time is about ten atomic vibration periods.

Stationary temperature tensor remains anisotropic in harmonic crystals.

Energy difference oscillates with decaying amplitude inversely proportional to time.

Abstract

We consider two high-frequency thermal processes in uniformly heated harmonic crystals relaxing towards equilibrium: (i) equilibration of kinetic and potential energies and (ii) redistribution of energy among spatial directions. Equation describing these processes with deterministic initial conditions is derived. Solution of the equation shows that characteristic time of these processes is of the order of ten periods of atomic vibrations. After that time the system practically reaches the stationary state. It is shown analytically that in harmonic crystals temperature tensor is not isotropic even in the stationary state. As an example, harmonic triangular lattice is considered. Simple formula relating the stationary value of the temperature tensor and initial conditions is derived. The function describing equilibration of kinetic and potential energies is obtained. It is shown that the difference between the energies (Lagrangian) oscillates around zero. Amplitude of these oscillations decays inversely proportional to time. Analytical results are in a good agreement with numerical simulations. Keywords: tensor temperature; nonequilibrium processes; transition to equilibrium; harmonic crystals; triangular lattice.