Abstract:
We investigate theoretically dark resonance spectroscopy for a dilute atomic vapor confined in a thin (micro-metric) cell. We identify the physical parameters characterizing the spectra and study their influence. We focus on a Hanle-type situation, with an optical irradiation under normal incidence and resonant with the atomic transition. The dark resonance spectrum is predicted to combine broad wings with a sharp maximum at line-center, that can be singled out when detecting a derivative of the dark resonance spectrum. This narrow signal derivative, shown to broaden only sub-linearly with the cell length, is a signature of the contribution of atoms slow enough to fly between the cell windows in a time as long as the characteristic ground state optical pumping time. We suggest that this dark resonance spectroscopy in micro-metric thin cells could be a suitable tool for probing the effective velocity distribution in the thin cell arising from the atomic desorption processes, and notably to identify the limiting factors affecting desorption under a grazing incidence.

Abstract:
Reading and manipulating the spin status of single magnetic molecules is of paramount importance both for applicative and fundamental purposes. The possibility to combine Electron Spin Resonance (ESR) and Scanning Tunnelling Microscopy (STM) has been explored one decade ago. A few experiments have raised the question whether or not an EPR spectrum of single molecule is detachable. Only a few data have been reported in modern surface science literature. To date it has yet to be proven to which extent ESR can be reliably and reproducibly performed on single molecules. We are setting up a new ESR-STM (spectrometer) to verify and study the effect of spin-spin correlations in the frequency spectrum of the tunneling current which flows through a single magnetic adsorbate and a metal surface. Here we discuss, the major experimental challenges that we are attempting to overcome.

Abstract:
It is proven that, under some conditions on $f$, the non-compact flat billiard $\Omega = \{ (x,y) \in \R_0^{+} \times \R_0^{+};\ 0\le y \le f(x) \}$ has no orbits going {\em directly} to $+\infty$. The relevance of such sufficient conditions is discussed.

Abstract:
It is proved that the quantization of the Volkovyski-Sinai model of ideal gas (in the Maxwell-Boltzmann statistics) enjoys at the thermodynamical limit the properties of mixing and ergodicity with respect to the quantum canonical Gibbs state. Plus, the average over the quantum state of a pseudo-differential operator is exactly the average over the classical canonical measure of its Weyl symbol.

Abstract:
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.

Abstract:
Let $f: [0, +\infty) \to (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincar\'e section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems.

Abstract:
In the scope of the statistical description of dynamical systems, one of the defining features of chaos is the tendency of a system to lose memory of its initial conditions (more precisely, of the distribution of its initial conditions). For a dynamical system preserving a probability measure, this property is named `mixing' and is equivalent to the decay of correlations for observables in phase space. For the class of dynamical systems preserving infinite measures, this probabilistic connection is lost and no completely satisfactory definition has yet been found which expresses the idea of losing track of the initial state of a system due to its chaotic dynamics. This is actually on open problem in the field of infinite ergodic theory. Virtually all the definitions that have been attempted so far use "local observables", that is, functions that essentially only "see" finite portions of the phase space. In this note we introduce the concept of "global observable", a function that gauges a certain quantity throughout the phase space. This concept is based on the notion of infinite-volume average, which plays the role of the expected value of a global observable. Endowed with these notions, whose rigorous definition is to be specified on a case-by-case basis, we give a number of definitions of infinite mixing. These fall in two categories: global-global mixing, which expresses the "decorrelation" of two global observables, and global-local mixing, where a global and a local observable are considered instead. These definitions are tested on two types of infinite-measure-preserving dynamical systems, the random walks and the Farey map.

Abstract:
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.

Abstract:
We explore the consequences of exactness or K-mixing on the notions of mixing (a.k.a. infinite-volume mixing) recently devised by the author for infinite-measure-preserving dynamical systems.

Abstract:
We prove the annealed Central Limit Theorem for random walks in bistochastic random environments on $Z^d$ with zero local drift. The proof is based on a "dynamicist's interpretation" of the system, and requires a much weaker condition than the customary uniform ellipticity. Moreover, recurrence is derived for $d \le 2$.