420 research outputs found
Chaotic dynamics in a quantum Fermi-Pasta-Ulam problem
We investigate the emergence of chaotic dynamics in a quantum Fermi - Pasta -
Ulam problem for anharmonic vibrations in atomic chains applying
semi-quantitative analysis of resonant interactions complemented by exact
diagonalization numerical studies. The crossover energy separating chaotic high
energy phase and localized (integrable) low energy phase is estimated. It
decreases inversely proportionally to the number of atoms until approaching the
quantum regime where this dependence saturates. The chaotic behavior appears at
lower energies in systems with free or fixed ends boundary conditions compared
to periodic systems. The applications of the theory to realistic molecules are
discussed.Comment: Submitted to Entrop
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
Thermostatistics in the neighborhood of the -mode solution for the Fermi-Pasta-Ulam system: from weak to strong chaos
We consider a -mode solution of the Fermi-Pasta-Ulam system. By
perturbing it, we study the system as a function of the energy density from a
regime where the solution is stable to a regime, where is unstable, first
weakly and then strongly chaotic. We introduce, as indicator of stochasticity,
the ratio (when is defined) between the second and the first moment of a
given probability distribution. We will show numerically that the transition
between weak and strong chaos can be interpreted as the symmetry breaking of a
set of suitable dynamical variables. Moreover, we show that in the region of
weak chaos there is numerical evidence that the thermostatistic is governed by
the Tsallis distribution.Comment: 15 pages, 5 figure
Dynamical thermalization of Bose-Einstein condensate in Bunimovich stadium
We study numerically the wavefunction evolution of a Bose-Einstein condensate
in a Bunimovich stadium billiard being governed by the Gross-Pitaevskii
equation. We show that for a moderate nonlinearity, above a certain threshold,
there is emergence of dynamical thermalization which leads to the Bose-Einstein
probability distribution over the linear eigenmodes of the stadium. This
distribution is drastically different from the energy equipartition over
oscillator degrees of freedom which would lead to the ultra-violet catastrophe.
We argue that this interesting phenomenon can be studied in cold atom
experiments.Comment: 6 pages, 6 figures. Accepted in Europhysics Letters. Video is
available at http://www.quantware.ups-tlse.fr/QWLIB/becstadium
Dynamics and thermalization of Bose-Einstein condensate in Sinai oscillator trap
We study numerically the evolution of Bose-Einstein condensate in the Sinai
oscillator trap described by the Gross-Pitaevskii equation in two dimensions.
In the absence of interactions this trap mimics the properties of Sinai
billiards where the classical dynamics is chaotic and the quantum evolution is
described by generic properties of quantum chaos and random matrix theory. We
show that, above a certain border, the nonlinear interactions between atoms
lead to the emergence of dynamical thermalization which generates the
statistical Bose-Einstein distribution over eigenmodes of the system without
interactions. Below the thermalization border the evolution remains
quasi-integrable. Such a Sinai oscillator trap, formed by the oscillator
potential and a repulsive disk located in the vicinity of the center, had been
already realized in rst experiments with the Bose-Einstein condensate formation
by Ketterle group in 1995 and we argue that it can form a convenient test bed
for experimental investigations of dynamical of thermalization. Possible links
and implications for Kolmogorov turbulence in absence of noise are also
discussed.Comment: 11 pages, 14 figures. Final version. Accepted forpublication at Phys.
Rev. A. Additional information available at
http://www.quantware.ups-tlse.fr/QWLIB/sinaioscillator
Many-body symbolic dynamics of a classical oscillator chain
We study a certain type of the celebrated Fermi-Pasta-Ulam particle chain,
namely the inverted FPU model, where the inter-particle potential has a form of
a quartic double well. Numerical evidence is given in support of a simple
symbolic description of dynamics (in the regime of sufficiently high potential
barrier between the wells) in terms of an (approximate) Markov process. The
corresponding transition matrix is formally identical to a ferromagnetic
Heisenberg quantum spin-1/2 chain with long range coupling, whose
diagonalization yields accurate estimates for a class of time correlation
functions of the model.Comment: 22 pages in LaTeX with 14 figures; submitted to Nonlinearity ;
corrected page offset proble
Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems
We investigate the connection between local and global dynamics of two
N-degree of freedom Hamiltonian systems with different origins describing
one-dimensional nonlinear lattices: The Fermi-Pasta-Ulam (FPU) model and a
discretized version of the nonlinear Schrodinger equation related to
Bose-Einstein Condensation (BEC). We study solutions starting in the vicinity
of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase
motion (OPM), which are known in closed form and whose linear stability can be
analyzed exactly. Our results verify that as the energy E increases for fixed
N, beyond the destabilization threshold of these orbits, all positive Lyapunov
exponents exhibit a transition between two power laws, occurring at the same
value of E. The destabilization energy E_c per particle goes to zero as N goes
to infinity following a simple power-law. However, using SALI, a very efficient
indicator we have recently introduced for distinguishing order from chaos, we
find that the two Hamiltonians have very different dynamics near their stable
SPOs: For example, in the case of the FPU system, as the energy increases for
fixed N, the islands of stability around the OPM decrease in size, the orbit
destabilizes through period-doubling bifurcation and its eigenvalues move
steadily away from -1, while for the BEC model the OPM has islands around it
which grow in size before it bifurcates through symmetry breaking, while its
real eigenvalues return to +1 at very high energies. Still, when calculating
Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov
exponents decrease following an exponential law and yield extensive
Kolmogorov--Sinai entropies per particle, in the thermodynamic limit of fixed
energy density E/N with E and N arbitrarily large.Comment: 29 pages, 10 figures, published at International Journal of
Bifurcation and Chaos (IJBC
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
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