# Thermal equilibration in a one-dimensional damped harmonic crystal

## TL;DR

This paper investigates how external damping affects thermal equilibration in a one-dimensional harmonic crystal, revealing a two-stage energy decay process with specific asymptotic behaviors for kinetic and potential energies.

## Contribution

It provides an analytical description of the damping influence on energy oscillations and decay in a harmonic crystal, supported by numerical verification.

## Key findings

- Energy oscillations decay exponentially in the underdamped case
- At large times, potential energy dominates with a $t^{-3/2}$ decay
- Kinetic energy decays as $t^{-5/2}$ at large times

## Abstract

The features for the unsteady process of thermal equilibration ("the fast motions") in a one-dimensional harmonic crystal lying in a viscous environment (e.g., a gas) are under investigation. It is assumed that initially the displacements of all the particles are zero and the particle velocities are random quantities with zero mean and a constant variance, thus, the system is far away from the thermal equilibrium. It is known that in the framework of the corresponding conservative problem the kinetic and potential energies oscillate and approach the equilibrium value that equals a half of the initial value of the kinetic energy. We show that the presence of the external damping qualitatively changes the features of this process. The unsteady process generally has two stages. At the first stage oscillations of kinetic and potential energies with decreasing amplitude, subjected to exponential decay, can be observed (this stage exists only in the underdamped case). At the second stage (which always exists), the oscillations vanish, and the energies are subjected to a power decay. The large-time asymptotics for the energy is proportional to $t^{-3/2}$ in the case of the potential energy and to $t^{-5/2}$ in the case the kinetic energy. Hence, at large values of time the total energy of the crystal is mostly the potential energy. The obtained analytic results are verified by independent numerical calculations.