10 research outputs found
Kolmogorov turbulence, Anderson localization and KAM integrability
The conditions for emergence of Kolmogorov turbulence, and related weak wave
turbulence, in finite size systems are analyzed by analytical methods and
numerical simulations of simple models. The analogy between Kolmogorov energy
flow from large to small spacial scales and conductivity in disordered solid
state systems is proposed. It is argued that the Anderson localization can stop
such an energy flow. The effects of nonlinear wave interactions on such a
localization are analyzed. The results obtained for finite size system models
show the existence of an effective chaos border between the
Kolmogorov-Arnold-Moser (KAM) integrability at weak nonlinearity, when energy
does not flow to small scales, and developed chaos regime emerging above this
border with the Kolmogorov turbulent energy flow from large to small scales.Comment: 8 pages, 6 figs, EPJB style
Resonance eigenfunction hypothesis for chaotic systems
A hypothesis about the average phase-space distribution of resonance
eigenfunctions in chaotic systems with escape through an opening is proposed.
Eigenfunctions with decay rate are described by a classical measure
that is conditionally invariant with classical decay rate and
is uniformly distributed on sets with the same temporal distance to the
quantum resolved chaotic saddle. This explains the localization of
fast-decaying resonance eigenfunctions classically. It is found to occur in the
phase-space region having the largest distance to the chaotic saddle. We
discuss the dependence on the decay rate and the semiclassical limit.
The hypothesis is numerically demonstrated for the standard map
Incommensurate standard map
We introduce and study the extension of the Chirikov standard map when the kick potential has two and three incommensurate spatial harmonics. This system is called the incommensurate standard map. At small kick amplitudes, the dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces, whereas above a certain kick strength, it becomes unbounded and diffusive. The quantum evolution at small quantum kick amplitudes is somewhat similar to the case of the Aubru-AndrĂ© model studied in mathematics and experiments with cold atoms in a static incommensurate potential. We show that for the quantum map there is also a metalinsulator transition in space whereas in momentum we have localization similar to the case of two-dimensional Anderson localization. In the case of three incommensurate frequencies of the space potential, the quantum evolution is characterized by the Anderson transition similar to the three-dimensional case of the disordered potential. We discuss possible physical systems with such a map description including dynamics of comets and dark matter in planetary systems.Fil: Ermann, Leonardo. ComisiĂłn Nacional de EnergĂa AtĂłmica; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Shepelyansky, Dima L.. UniversitĂ© de Toulouse; Franci
Self-duality triggered dynamical transition
A basic result about the dynamics of spinless quantum systems is that the
Maryland model exhibits dynamical localization in any dimension. Here we
implement mathematical spectral theory and numerical experiments to show that
this result does not hold, when the 2-dimensional Maryland model is endowed
with spin 1/2 -- hereafter dubbed spin-Maryland (SM) model. Instead, in a
family of SM models, tuning the (effective) Planck constant drives dynamical
localization{delocalization transitions of topological nature. These
transitions are triggered by the self-duality, a symmetry generated by some
transformation in the parameter -- the inverse Planck constant -- space. This
provides significant insights to new dynamical phenomena such as what occur in
the spinful quantum kicked rotor.Comment: 18 pages, 6 figure
Arithmetic Phase Transitions For Mosaic Maryland Model
We give a precise description of spectral types of the Mosaic Maryland model
with any irrational frequency, which provides a quasi-periodic unbounded model
with non-monotone potential has arithmetic phase transition.Comment: arXiv admin note: substantial text overlap with arXiv:2205.04021 by
other author
Chaotic Einstein-Podolsky-Rosen pairs, measurements and time reversal
We consider a situation when evolution of an entangled
Einstein-Podolsky-Rosen (EPR) pair takes place in a regime of quantum chaos
being chaotic in the classical limit. This situation is studied on an example
of chaotic pair dynamics described by the quantum Chirikov standard map. The
time evolution is reversible even if a presence of small errors breaks time
reversal of classical dynamics due to exponential growth of errors induced by
exponential chaos instability. However, the quantum evolution remains
reversible since a quantum dynamics instability exists only on a
logarithmically short Ehrenfest time scale. We show that due to EPR pair
entanglement a measurement of one particle at the moment of time reversal
breaks exact time reversal of another particle which demonstrates only an
approximate time reversibility. This result is interpreted in the framework of
the Schmidt decomposition and Feynman path integral formulation of quantum
mechanics. The time reversal in this system has already been realized with cold
atoms in kicked optical lattices in absence of entanglement and measurements.
On the basis of the obtained results we argue that the experimental
investigations of time reversal of chaotic EPR pairs is within reach of present
cold atom capabilities.Comment: 15 pages, 19 figure
Dynamical thermalization of interacting fermionic atoms in a sinai oscillator trap
We study numerically the problem of dynamical thermalization of interacting cold fermionic atoms placed in an isolated Sinai oscillator trap. This system is characterized by a quantum chaos regime for one-particle dynamics. We show that, for a many-body system of cold atoms, the interactions, with a strength above a certain quantum chaos border given by the Ă…berg criterion, lead to the Fermi–Dirac distribution and relaxation of many-body initial states to the thermalized state in the absence of any contact with a thermostate. We discuss the properties of this dynamical thermalization and its links with the Loschmidt–Boltzmann dispute.Fil: Frahm, Klaus M.. UniversitĂ© Paul Sabatier; Francia. Centre National de la Recherche Scientifique; FranciaFil: Ermann, Leonardo. ComisiĂłn Nacional de EnergĂa AtĂłmica; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Shepelyansky, Dima L.. UniversitĂ© Paul Sabatier; Franci
Many Body Quantum Chaos
This editorial remembers Shmuel Fishman, one of the founding fathers of the research field "quantum chaos", and puts into context his contributions to the scientific community with respect to the twelve papers that form the special issue
International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022
Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022.
Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress.
The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library