Adiabatic invariant at the thermodynamic limit in nonlinear lattices
speaker: Simone Paleari (Università degli Studi di Milano - Dipartimento di Matematica)abstract: A natural source of ergodicity breaking over long albeit finite time scale is given by the presence of a quasi conserved quantity. This in turn is a call for perturbation methods; unfortunately, it is a well known limit of the classical results of this theory (like KAM or Nekhoroshev theorem) to suffer a bad dependence on the number of degrees of freedom, often resulting in void or non applicable statements in the thermodynamic limit.
It is thus an important challenge in Hamiltonian dynamics the development of a perturbation theory for Hamiltonian systems with an arbitrarily large number of degrees of freedom, and in particular in the thermodynamic limit. Indeed, motivated by the problems arising in the foundations of Statistical Mechanics, it is relevant to consider also large systems with non vanishing energy per particle (which corresponds to a non zero temperature in the physical model).
I will try to present some results in the direction of the aforementioned goals. The model considered are nonlinear chains, in particolar finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions.
We construct an extensive adiabatic invariant in the thermodynamic limit. Given a fixed and sufficiently small value of the coupling constant $a$, the evolution of the adiabatic invariant is controlled up to times scaling as $\beta{1a}$ for any large enough value of the inverse temperature $\beta$. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbation constructions and the subsequent estimates are consistent with the extensive nature of the system.
timetable:
Thu 6 Jun, 16:20 - 17:05, Aula Dini
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