Marco Lenci

Large deviations for ideal quantum systems

We consider a general d-dimensional quantum system of non-interacting particles in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain Λ of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of Λ, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.

Typicality of recurrence for Lorentz gases

It is a safe conjecture that most (not necessarily periodic) two-dimensional Lorentz gases with finite horizon are recurrent. Here we formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which i.i.d. random scatterers are placed in each cell of a co-compact lattice in the plane. We prove that the typical Lorentz gas, in the sense of Baire, is recurrent, and give results in the direction of showing that recurrence is an almost sure property (including a zero-one law that holds in every dimension). A few toy models illustrate the extent of these results.

Aperiodic Lorentz gas: recurrence and ergodicity

We prove that any generic (i.e., possibly aperiodic) Lorenz gas in two dimensions, with finite horizon and non-degenerate geometrical features, is ergodic if it is recurrent. We also give examples of aperiodic recurrent gases.

On Infinite-Volume Mixing

In the context of the long-standing issue of mixing in infinite ergodic theory, we introduce the idea of mixing for observables possessing an infinite-volume average. The idea is borrowed from statistical mechanics and appears to be relevant, at least for extended systems with a direct physical interpretation. We discuss the pros and cons of a few mathematical definitions that can be devised, testing them on a prototypical class of infinite measure-preserving dynamical systems, namely, the random walks.

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a striplike set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called quenched random Lorentz tube. We prove that under general conditions, almost every system in the ensemble is recurrent.

An infinite step billiard

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain , with elastic reflections on the boundary; here , and . After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example . Playing an important role in this case are the so-called escape orbits, that is, orbits going to monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard.

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards.

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of superdiffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time range accessible to numerical simulations. We compare our simulations to analytical results for the variance of the anomalously rescaled limiting normal distributions.

Escape Orbits for Non-Compact Flat Billiards

It is proven that, under some conditions on f, the non-compact flat billiard Omega={(x,y) in R(0) (+)xR(0) (+); 0</=y</=f(x)} has no orbits going directly to + infinity. The relevance of such sufficient conditions is discussed. (c) 1996 American Institute of Physics.

Random walks in random environments without ellipticity

We consider random walks in random environments on Zd. Under a transitivity hypothesis that is much weaker than the customary ellipticity condition, and assuming an absolutely continuous invariant measure on the space of the environments, we prove the ergodicity of the annealed process w.r.t. the dynamics “from the point of view of the particle”. This implies in particular that the environment viewed from the particle is ergodic. As an example of application of this result, we give a general form of the quenched Invariance Principle for walks in doubly stochastic environments with zero local drift (martingale condition).

Hyperbolic billiards with nearly flat focusing boundaries, I

Abstract The standard Wojtkowski–Markarian–Donnay–Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms.