Selfdecomposability and selfsimilarity: a concise primer

We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.Comment: 24 pages, 3 figures; corrected misprint in the title; redactional modifications required by the referee; added references from [16] to [28];. Accepted and in press on Physica

Thou shalt not say "at random" in vain: Bertrand's paradox exposed

We review the well known Bertrand paradoxes, and we first maintain that they do not point to any probabilistic inconsistency, but rather to the risks incurred with a careless use of the locution "at random". We claim then that these paradoxes spring up also in the discussion of the celebrated Buffon's needle problem, and that they are essentially related to the definition of (geometrical) probabilities on "uncountably" infinite sets. A few empirical remarks are finally added to underline the difference between "passive" and "active" randomness, and the prospects of any experimental decisionComment: 17 pages, 4 figures. Added: Appendix A; References 7, 8, 10; Modified: Abstract; Section 4; a few sentences elsewher

Controlled quantum evolutions and transitions

We study the nonstationary solutions of Fokker-Planck equations associated to either stationary or nonstationary quantum states. In particular we discuss the stationary states of quantum systems with singular velocity fields. We introduce a technique that allows to realize arbitrary evolutions ruled by these equations, to account for controlled quantum transitions. The method is illustrated by presenting the detailed treatment of the transition probabilities and of the controlling time-dependent potentials associated to the transitions between the stationary, the coherent, and the squeezed states of the harmonic oscillator. Possible extensions to anharmonic systems and mixed states are briefly discussed and assessed.Comment: 24 pages, 4 figure

Levy processes and Schroedinger equation

We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We consider a Levy-Schroedinger equation where the usual kinetic energy operator - the Laplacian - is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the fractional Schroedinger equation. A few examples are finally given and we find that there are physically relevant models (such as a form of the relativistic Schroedinger equation) that are in the domain of the non-stable, Levy-Schroedinger equations.Comment: 10 pages; changed the TeX documentclass; added references [21] and [22] and comments about them; changed definitions (11) and (12); added acknowledgments; small changes scattered in the tex

A stochastic-hydrodynamic model of halo formation in charged particle beams

The formation of the beam halo in charged particle accelerators is studied in the framework of a stochastic-hydrodynamic model for the collective motion of the particle beam. In such a stochastic-hydrodynamic theory the density and the phase of the charged beam obey a set of coupled nonlinear hydrodynamic equations with explicit time-reversal invariance. This leads to a linearized theory that describes the collective dynamics of the beam in terms of a classical Schr\"odinger equation. Taking into account space-charge effects, we derive a set of coupled nonlinear hydrodynamic equations. These equations define a collective dynamics of self-interacting systems much in the same spirit as in the Gross-Pitaevskii and Landau-Ginzburg theories of the collective dynamics for interacting quantum many-body systems. Self-consistent solutions of the dynamical equations lead to quasi-stationary beam configurations with enhanced transverse dispersion and transverse emittance growth. In the limit of a frozen space-charge core it is then possible to determine and study the properties of stationary, stable core-plus-halo beam distributions. In this scheme the possible reproduction of the halo after its elimination is a consequence of the stationarity of the transverse distribution which plays the role of an attractor for every other distribution.Comment: 18 pages, 20 figures, submitted to Phys. Rev. ST A

Levy-Student Distributions for Halos in Accelerator Beams

We describe the transverse beam distribution in particle accelerators within the controlled, stochastic dynamical scheme of the Stochastic Mechanics (SM) which produces time reversal invariant diffusion processes. This leads to a linearized theory summarized in a Shchr\"odinger--like (\Sl) equation. The space charge effects have been introduced in a recent paper~\cite{prstab} by coupling this \Sl equation with the Maxwell equations. We analyze the space charge effects to understand how the dynamics produces the actual beam distributions, and in particular we show how the stationary, self--consistent solutions are related to the (external, and space--charge) potentials both when we suppose that the external field is harmonic (\emph{constant focusing}), and when we \emph{a priori} prescribe the shape of the stationary solution. We then proceed to discuss a few new ideas~\cite{epac04} by introducing the generalized Student distributions, namely non--Gaussian, L\'evy \emph{infinitely divisible} (but not \emph{stable}) distributions. We will discuss this idea from two different standpoints: (a) first by supposing that the stationary distribution of our (Wiener powered) SM model is a Student distribution; (b) by supposing that our model is based on a (non--Gaussian) L\'evy process whose increments are Student distributed. We show that in the case (a) the longer tails of the power decay of the Student laws, and in the case (b) the discontinuities of the L\'evy--Student process can well account for the rare escape of particles from the beam core, and hence for the formation of a halo in intense beams.Comment: revtex4, 18 pages, 12 figure