On the approach to equilibrium of an Hamiltonian chain of anharmonic oscillators

In this note we study the approach to equilibrium of a chain of anharmonic oscillators. We find indications that a sufficiently large system always relaxes to the usual equilibrium distribution. There is no sign of an ergodicity threshold. The time however to arrive to equilibrium diverges when g ! 0, g being the anharmonicity. A debated issue is the approach to equilibrium of an Hamiltonian system. A well studied problem is a chain of anharmonic oscillators. The first numerical simulations have been done more than 40 years ago [1]: the authors found that for sufficiently small anharmonicity the system does not goes to the usual Boltzmann Gibbs equilibrium and there is strong memory of the initial conditions especially if the systems starts from a relatively smooth configurations, i.e. only long wave length modes are populated. In the case of a finite system the KAM theorem [2] states that for small anharmonicity g the behaviour of the system is not ergodic and there is threshold val..