Characterization of thermalized Fermi-Pasta-Ulam chains

Authors
Gershgorin, Boris
Description
The Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal equilibrium} are studied from both wave-interaction and particle-interaction points of view. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting \textit{renormalized} nonlinear waves. These waves possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized $\beta$-FPU chain is found to be in very good agreement with its numerically measured value. Moreover, we show that the dynamical scenario for thermalized $\beta$-FPU chains is spatially highly localized discrete breathers riding chaotically on spatially extended, renormalized waves. We present numerical evidence of existence of discrete breathers in thermal equilibrium.
Keywords
Mathematical Physics, Mathematics - Dynamical Systems