Stefano RUFFO

Dipartimento di Energetica
Via S. Marta 3 
50139 Firenze
Italie
ruffo@ingfi1.ing.unifi.it
 

Time scales for the relaxation to thermodynamical equilibrium.

Abstract: We have studied two physical examples where the time-scale for the relaxation to thermodynamical equilibrium of an isolated system started from a non-equilibrium initial condition becomes exceedingly large:

a) The Fermi-Pasta-Ulam (FPU) systems at small energy density e. Here a power-law increase of the time-scale as e- a (a \approx 3$ for the FPU-b system) is shown numerically and justified analytically. This sharp increase was previously interpreted as a ``freezing" of the non-equilibrium initial state (a packet of Fourier modes with the same energy), while a slow ``diffusion" to the energy equipartition state is now exhibited.

b) The Hamiltonian Mean-Field model. $N$ particles moving on a circle are all coupled via an attractive or repulsive cosine interaction. At equilibrium the attractive case has a clustering-declustering phase transition. When started in a ``water-bag" initial condition near the phase transition the systems stays far from equilibrium for a time which increses with $N$. The repulsive case also shows time-scales which increase with $N$ in the high- energy region. Both effects are related to a slow diffusion of the orbit in the phase-space, which remains to be fully characterized (perhaps some progress will be made before the workshop in Aussois).