103 research outputs found
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
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
Mass spectrum from stochastic Levy-Schroedinger relativistic equations: possible qualitative predictions in QCD
Starting from the relation between the kinetic energy of a free Levy-Schroedinger particle and the logarithmic characteristic of the underlying stochastic process, we show that it is possible to get a precise relation between renormalizable field theories and a specific Levy process. This subsequently leads to a particular cut-off in the perturbative diagrams and can produce a phenomenological mass spectrum that allows an interpretation of quarks and leptons distributed in the three families of the standard model
Mass spectrum and Lévy-Scrödinger relativistic equation
We introduce a modification in the relativistic hamiltonian in such a way that (1) the relativistic Schr\"odinger equations can always be based on an underlying L\'evy process, (2) several families of particles with different rest masses can be selected, and finally (3) the corresponding Feynman diagrams are convergent when we have at least three different masses
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 ,
where is the number of particles in the beam and 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 Schr\"odinger 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.Comment: 15 pages, 9 figure
L\'evy-Schr\"odinger wave packets
We analyze the time--dependent solutions of the pseudo--differential
L\'evy--Schr\"odinger wave equation in the free case, and we compare them with
the associated L\'evy processes. We list the principal laws used to describe
the time evolutions of both the L\'evy process densities, and the
L\'evy--Schr\"odinger wave packets. To have self--adjoint generators and
unitary evolutions we will consider only absolutely continuous, infinitely
divisible L\'evy noises with laws symmetric under change of sign of the
independent variable. We then show several examples of the characteristic
behavior of the L\'evy--Schr\"odinger wave packets, and in particular of the
bi-modality arising in their evolutions: a feature at variance with the typical
diffusive uni--modality of both the L\'evy process densities, and the usual
Schr\"odinger wave functions.Comment: 41 pages, 13 figures; paper substantially shortened, while keeping
intact examples and results; changed format from "report" to "article";
eliminated Appendices B, C, F (old names); shifted Chapters 4 and 5 (old
numbers) from text to Appendices C, D (new names); introduced connection
between Relativistic q.m. laws and Generalized Hyperbolic law
Controlled quantum evolutions and stochastic mechanics
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
On some results of Cufaro Petroni about Student t-processes
This paper deals with Student t-processes as studied in (Cufaro Petroni N
2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend some
conjectures expressed by Cufaro Petroni about the asymptotical behavior of a
Student t-process and the expansion of its density. First, the explicit
asymptotic behavior of any real positive convolution power of a Student
t-density with any real positive degrees of freedom is given in the
multivariate case; then the integer convolution power of a Student
t-distribution with odd degrees of freedom is shown to be a convex combination
of Student t-densities with odd degrees of freedom. At last, we show that this
result does not extend to the case of non-integer convolution powers.Comment: 11 page
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