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Seminar in Mathematical physics and related subjects: “Perturbation theory for Hamiltonian systems with infinitely many degrees of freedom: a probabilistic approach”

Thursday, March 21, 2019, 14:30 - Room 1BC45 - Alberto Maiocchi (Milano)

ARGOMENTI: Seminars

Alberto Maiocchi (Milano)
“Perturbation theory for Hamiltonian systems with infinitely many degrees of freedom: a probabilistic approach”
Thursday, March 21, 2019, 14:30
Room 1BC45

Abstract
We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered systems in a surprising fashion. For this aim use is made of perturbation theory techniques such as KAM theorem or Nekhoroshev theorem, but they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We cite some results on infinite chains of interacting particles and Hamiltonian partial differential equations, and we explain in details the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In the latter case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, both with analytical and numerical works, we show the advancements with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.