This research project consists of three parts: Part I deals with billiards on non-compact tables, as models for hyperbolic dynamical systems with an infinite invariant measure. The plan is to extend to these dynamical systems certain important results available for compact billiards, including a suitable version of Pesin's Theory. This will require the rewriting of some notions of ergodic theory for the case of a non-probability measure. Part II concerns random billiards; more precisely, billiards with a random table (changing after each collision), a quenched random table (selected once and for all from a random ensemble), or a random law of reflection (in a fixed table). In the first two cases there is a physical invariant measure, and the PI will study its ergodic properties (including Lyapunov exponents and the like) for a typical realization of the random process. In the third case an equilibrium measure can be shown to exist, and its asymptotic properties will be investigated, together with the natural question of the stochastic stability at the zero-noise limit. Similar perturbative questions are considered for the other systems as well. Part III, in the realm of equilibrium statistical mechanics, considers the question of quantum large deviations. In a collaborative project, the PI sets out to initiate a theory of large deviations for a noncommutative quasi-local algebra on a d-dimensional lattice. Of primary interest is the convergence of the moment-generating function for an extensive observable, and the smoothness and physical significance of its limit.
Non-compact, random billiard systems; Quantum large deviations
LENCI, MARCO
2007
Abstract
This research project consists of three parts: Part I deals with billiards on non-compact tables, as models for hyperbolic dynamical systems with an infinite invariant measure. The plan is to extend to these dynamical systems certain important results available for compact billiards, including a suitable version of Pesin's Theory. This will require the rewriting of some notions of ergodic theory for the case of a non-probability measure. Part II concerns random billiards; more precisely, billiards with a random table (changing after each collision), a quenched random table (selected once and for all from a random ensemble), or a random law of reflection (in a fixed table). In the first two cases there is a physical invariant measure, and the PI will study its ergodic properties (including Lyapunov exponents and the like) for a typical realization of the random process. In the third case an equilibrium measure can be shown to exist, and its asymptotic properties will be investigated, together with the natural question of the stochastic stability at the zero-noise limit. Similar perturbative questions are considered for the other systems as well. Part III, in the realm of equilibrium statistical mechanics, considers the question of quantum large deviations. In a collaborative project, the PI sets out to initiate a theory of large deviations for a noncommutative quasi-local algebra on a d-dimensional lattice. Of primary interest is the convergence of the moment-generating function for an extensive observable, and the smoothness and physical significance of its limit.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
