Prethermalization in a classical phonon field: Slow relaxation of the number of phonons

François Huveneers,Jani Lukkarinen

doi:10.1103/physrevresearch.2.022034

Abstract

We investigate the emergence of an astonishingly long pre-thermal plateau in a classical phonon field, here a harmonic chain with on-site pinning. Integrability is broken by a weak anharmonic on-site potential with strength $\lambda$. In the small $\lambda$ limit, the approach to equilibrium of a translation invariant initial state is described by kinetic theory. However, when the phonon band becomes narrow, we find that the (non-conserved) number of phonons relaxes on much longer time scales than kinetic. We establish rigorous bounds on the relaxation time, and develop a theory that yields exact predictions for the dissipation rate in the limit $\lambda \to 0$. We compare the theoretical predictions with data from molecular dynamics simulations and find good agreement. Our work shows how classical systems may exhibit phenomena which at the first glance appear to require quantization.

Highlights

Thermalization is one of the most commonly encountered physical phenomena, and yet, it still remains poorly understood
The proper equilibrium is only reached after a longer time τ2, scaling as τ2 ∼ λ−2p for some p 1 which becomes arbitrarily large as the width of the band narrows to zero
If the system is prepared in a translation invariant state with zero average, after a short transient time, it evolves towards a generalized Gibbs ensemble (GGE) characterized by a Wigner function W, i.e., a Gaussian state exp [− BZ dk n(k)/W (k)]/Z

Summary

Rapid Communications

Prethermalization in a classical phonon field: Slow relaxation of the number of phonons. We investigate the emergence of an astonishingly long prethermal plateau in a classical phonon field, here, a harmonic chain with on-site pinning. In the small λ limit, the approach to equilibrium of a translation invariant initial state is described by kinetic theory. When the phonon band becomes narrow, we find that the (nonconserved) number of phonons relaxes on much longer timescales than kinetic. We establish rigorous bounds on the relaxation time and develop a theory that yields exact predictions for the dissipation rate in the limit λ → 0. We compare the theoretical predictions with data from molecular-dynamics simulations and find good agreement. Our Rapid Communication shows how classical systems may exhibit phenomena which, at the first glance, appear to require quantization

Introduction

FRANÇOIS HUVENEERS AND JANI LUKKARINEN

Gn such that

Conclusions and outlook