Numerical experiments on the modulation theory for the nonlinear atomic chain

Modulation theory with periodic travelling waves is a powerful, but not rigorous tool to derive a thermodynamic description for atomic chains with nearest neighbour interactions (FPU chains). This theory is sufficiently complex to deal with strong oscillations on the microscopic scale, and therefore it is capable to describe the creation of temperature and the transport of heat on a macroscopic scale. In this paper we investigate the validity of modulation theory by means of several numerical experiments. We start with a survey on the foundations of modulation theory. In particular, we discuss the hyperbolic scaling, the notion of cold data, microscopic oscillations and Young measures, periodic and modulated travelling waves, and, finally, the resulting macroscopic conservation laws. Afterwards we discuss how the validity of a macroscopic theory may be tested within numerical simulations of the microscopic dynamics. To this end we describe an approach to thermodynamic data exploration which is motivated by the theory of Young measures, and relies on mesoscopic windows in space and time. The last part is devoted to several numerical experiments including examples with periodic boundary conditions and smooth initial data, and macroscopic Riemann problems. We interpret the outcome of these experiments in the framework of thermodynamics, and end up with two conclusions. (1) There are many examples for which modulation theory provides in fact the right thermodynamic description because it can predict both the structure of the microscopic oscillations and their macroscopic evolution correctly. (2) Modulation theory will fail if the oscillations exhibit a more complicate structure.