Micro–macro transition in the atomic chain via Whitham's modulation equation

Abstract

The subject matter of this paper is the thermodynamic description of the nonlinear atomic chain with temperature. For this reason we consider special approximate solutions of Newton's equations, in which the atoms perform microscopic oscillations in the form of modulated travelling waves. We start with an existence result for periodic travelling waves with arbitrary large amplitudes and study several examples including the harmonic chain, the hard sphere model and the small-amplitude approximation. Then we discuss the thermodynamic properties of travelling waves and derive the corresponding Gibbs equation. Afterwards we focus on the macroscopic evolution of modulated travelling waves. For this purpose we apply Whitham's modulation theory to the atomic chain and derive the modulation equation, which turns out to be a system of four macroscopic conservation laws. The last part is devoted to the justification problem: we state a conjecture for the general case and prove this conjecture for the harmonic chain and the hard sphere model.