Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

Abstact:

The numerical experiment of Fermi, Pasta and Ulam in 1954 aimed to probe
ergodicity in an one-dimensional chain of N weakly nonlinearly coupled oscillators, 
however led to an unexpected integrable-like behaviour. In the present talk I 
will discuss and compare the stages of dynamics in the Fermi-Pasta-Ulam (FPU) 
model for different classes of initial conditions, as probed by the first Toda integral J. 
In general, we find sigmoid curves, whose behavior is analogous to observables 
like the spectral entropy and correlation functions, but without various integrable 
effects contributing fluctuations in the observable's temporal progression. We also 
find that the Toda integral J accurately determines the slow diffusion at which an 
FPU trajectory wanders transversally to the 'Toda tori' and clearly determines the 
two fundamental timescales, namely: i) the time of stability, where FPU acts like 
Toda, and ii) the time to equilibrium.