arXiv subjects

Nicola Cufaro Petroni

Publications and source records attributed to Nicola Cufaro Petroni.

24 records · Page 2Linked to original sources

arXiv2008

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.

cond-mat.stat-mech↗

arXiv2007

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.

cond-mat.stat-mech↗

arXiv2007

Mixtures in non stable Levy processes

We analyze the Levy processes produced by means of two interconnected classes of non stable, infinitely divisible distribution: the Variance Gamma and the Student laws. While the Variance Gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that -- at least for one particular type of Student processes suggested by recent empirical results, and for integral times -- the distribution of the process is a mixture of other types of Student distributions, randomized by means of a new probability distribution. The mixture is such that along the time the asymptotic behavior of the probability density functions always coincide with that of the generating Student law. We put forward the conjecture that this can be a general feature of the Student processes. We finally analyze the Ornstein--Uhlenbeck process driven by our Levy noises and show a few simulation of it.

math.PR↗

arXiv2003

Stochastic collective dynamics of charged--particle beams in the stability regime

We introduce a description of the collective transverse dynamics of charged (proton) beams in the stability regime by suitable classical stochastic fluctuations. In this scheme, the collective beam dynamics is described by time--reversal invariant diffusion processes deduced by stochastic variational principles (Nelson processes). By general arguments, we show that the diffusion coefficient, expressed in units of length, is given by $λ_c\sqrt{N}$, where $N$ is the number of particles in the beam and $λ_c$ the Compton wavelength of a single constituent. This diffusion coefficient represents an effective unit of beam emittance. The hydrodynamic equations of the stochastic dynamics can be easily recast in the form of a Schrödinger equation, with the unit of emittance replacing the Planck action constant. This fact provides a natural connection to the so--called ``quantum--like approaches'' to beam dynamics. The transition probabilities associated to Nelson processes can be exploited to model evolutions suitable to control the transverse beam dynamics. In particular we show how to control, in the quadrupole approximation to the beam--field interaction, both the focusing and the transverse oscillations of the beam, either together or independently.

cond-mat.stat-mech↗

arXiv2003

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 Schrödinger 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.

cond-mat.stat-mech↗

arXiv1999

Controlled quantum evolutions

We perform a detailed analysis of the non stationary solutions of the evolution (Fokker-Planck) equations associated to either stationary or non stationary quantum states by the stochastic mechanics. For the excited stationary states of quantum systems with singular velocity fields we explicitely discuss the exact solutions for the HO case. Moreover the possibility of modifying the original potentials in order to implement arbitrary evolutions ruled by these equations is discussed with respect to both possible models for quantum measurements and applications to the control of particle beams in accelerators.

quant-ph↗