Three-wave resonant interactions in the diatomic chain with cubic anharmonic potential: theory and simulations

TL;DR

This paper investigates three-wave resonant interactions in a diatomic chain with cubic anharmonic potential, deriving wave kinetic equations and demonstrating energy transfer and thermalization through theoretical analysis and simulations.

Contribution

It introduces a novel wave turbulence framework for diatomic chains, revealing three-wave resonances and faster thermalization compared to monoatomic chains.

Findings

Resonances occur only if heavy/light mass ratio is less than 3.

Energy transfer is governed by three-wave interactions in diatomic chains.

Numerical simulations confirm theoretical predictions of thermalization dynamics.

Abstract

We consider a diatomic chain characterized by a cubic anharmonic potential. After diagonalizing the harmonic case, we study in the new canonical variables, the nonlinear interactions between the acoustical and optical branches of the dispersion relation. Using the {\it wave turbulence} approach, we formally derive two coupled wave kinetic equations, each describing the evolution of the wave action spectral density associated to each branch. An $H$-theorem shows that there exist an irreversible transfer of energy that leads to an equilibrium solution characterized by the equipartition of energy in the new variables. While in the monoatomic cubic chain, in the large box limit, the main nonlinear transfer mechanism is based on four-wave resonant interactions, the diatomic one is ruled by a three wave resonant process (two acoustical and one optical wave): thermalization happens on shorter time scale for the diatomic chain with respect to the standard chain. Resonances are possible only if the ratio between the heavy and light masses is less than 3. Numerical simulations of the deterministic equations support our theoretical findings.