2,791 research outputs found
Non-trivial stably free modules over crossed products
We consider the class of crossed products of noetherian domains with
universal enveloping algebras of Lie algebras. For algebras from this class we
give a sufficient condition for the existence of projective non-free modules.
This class includes Weyl algebras and universal envelopings of Lie algebras,
for which this question, known as noncommutative Serre's problem, was
extensively studied before. It turns out that the method of lifting of
non-trivial stably free modules from simple Ore extensions can be applied to
crossed products after an appropriate choice of filtration. The motivating
examples of crossed products are provided by the class of RIT algebras,
originating in non-equilibrium physics.Comment: 13 page
One-Dimensional Discrete Stark Hamiltonian and Resonance Scattering by Impurities
A one-dimensional discrete Stark Hamiltonian with a continuous electric field
is constructed by extension theory methods. In absence of the impurities the
model is proved to be exactly solvable, the spectrum is shown to be simple,
continuous, filling the real axis; the eigenfunctions, the resolvent and the
spectral measure are constructed explicitly. For this (unperturbed) system the
resonance spectrum is shown to be empty. The model considering impurity in a
single node is also constructed using the operator extension theory methods.
The spectral analysis is performed and the dispersion equation for the
resolvent singularities is obtained. The resonance spectrum is shown to contain
infinite discrete set of resonances. One-to-one correspondence of the
constructed Hamiltonian to some Lee-Friedrichs model is established.Comment: 20 pages, Latex, no figure
Criticality, Fractality and Intermittency in Strong Interactions
Assuming a second-order phase transition for the hadronization process, we
attempt to associate intermittency patterns in high-energy hadronic collisions
to fractal structures in configuration space and corresponding intermittency
indices to the isothermal critical exponent at the transition temperature. In
this approach, the most general multidimensional intermittency pattern,
associated to a second-order phase transition of the strongly interacting
system, is determined, and its relevance to present and future experiments is
discussed.Comment: 15 pages + 2 figures (available on request), CERN-TH.6990/93,
UA/NPPS-5-9
Guest editorial: Scientific seminar of the Italian Association of Transport Academicians (SIDT) 2019
The Fractal Geometry of Critical Systems
We investigate the geometry of a critical system undergoing a second order
thermal phase transition. Using a local description for the dynamics
characterizing the system at the critical point T=Tc, we reveal the formation
of clusters with fractal geometry, where the term cluster is used to describe
regions with a nonvanishing value of the order parameter. We show that,
treating the cluster as an open subsystem of the entire system, new
instanton-like configurations dominate the statistical mechanics of the
cluster. We study the dependence of the resulting fractal dimension on the
embedding dimension and the scaling properties (isothermal critical exponent)
of the system. Taking into account the finite size effects we are able to
calculate the size of the critical cluster in terms of the total size of the
system, the critical temperature and the effective coupling of the long
wavelength interaction at the critical point. We also show that the size of the
cluster has to be identified with the correlation length at criticality.
Finally, within the framework of the mean field approximation, we extend our
local considerations to obtain a global description of the system.Comment: 1 LaTeX file, 4 figures in ps-files. Accepted for publication in
Physical Review
Classical evolution of fractal measures on the lattice
We consider the classical evolution of a lattice of non-linear coupled
oscillators for a special case of initial conditions resembling the equilibrium
state of a macroscopic thermal system at the critical point. The displacements
of the oscillators define initially a fractal measure on the lattice associated
with the scaling properties of the order parameter fluctuations in the
corresponding critical system. Assuming a sudden symmetry breaking (quench),
leading to a change in the equilibrium position of each oscillator, we
investigate in some detail the deformation of the initial fractal geometry as
time evolves. In particular we show that traces of the critical fractal measure
can sustain for large times and we extract the properties of the chain which
determine the associated time-scales. Our analysis applies generally to
critical systems for which, after a slow developing phase where equilibrium
conditions are justified, a rapid evolution, induced by a sudden symmetry
breaking, emerges in time scales much shorter than the corresponding relaxation
or observation time. In particular, it can be used in the fireball evolution in
a heavy-ion collision experiment, where the QCD critical point emerges, or in
the study of evolving fractals of astrophysical and cosmological scales, and
may lead to determination of the initial critical properties of the Universe
through observations in the symmetry broken phase.Comment: 15 pages, 15 figures, version publiced at Physical Review
INTRINSIC MECHANISM FOR ENTROPY CHANGE IN CLASSICAL AND QUANTUM EVOLUTION
It is shown that the existence of a time operator in the Liouville space
representation of both classical and quantum evolution provides a mechanism for
effective entropy change of physical states. In particular, an initially
effectively pure state can evolve under the usual unitary evolution to an
effectively mixed state.Comment: 20 pages. For more information or comments contact E. Eisenberg at
[email protected] (internet)
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