289 research outputs found
Two stories outside Boltzmann-Gibbs statistics: Mori's q-phase transitions and glassy dynamics at the onset of chaos
First, we analyze trajectories inside the Feigenbaum attractor and obtain the
atypical weak sensitivity to initial conditions and loss of information
associated to their dynamics. We identify the Mori singularities in its
Lyapunov spectrum with the appearance of a special value for the entropic index
q of the Tsallis statistics. Secondly, the dynamics of iterates at the
noise-perturbed transition to chaos is shown to exhibit the characteristic
elements of the glass transition, e.g. two-step relaxation, aging, subdiffusion
and arrest. The properties of the bifurcation gap induced by the noise are seen
to be comparable to those of a supercooled liquid above a glass transition
temperature.Comment: Proceedings of: 31st Workshop of the International School of Solid
State Physics, Complexity, Metastability and Nonextensivity, Erice (Sicily)
20-26 July 2004 World Scientific in the special series of the E. Majorana
conferences, in pres
Multifractality and nonextensivity at the edge of chaos of unimodal maps
We examine both the dynamical and the multifractal properties at the chaos
threshold of logistic maps with general nonlinearity . First we determine
analytically the sensitivity to initial conditions . Then we consider
a renormalization group (RG) operation on the partition function of the
multifractal attractor that eliminates one half of the multifractal points each
time it is applied. Invariance of fixes a length-scale transformation
factor in terms of the generalized dimensions . There
exists a gap in the values of equal to where is the
-generalized Lyapunov exponent and is the nonextensive entropic index.
We provide an interpretation for this relationship - previously derived by Lyra
and Tsallis - between dynamical and geometrical properties. Key Words: Edge of
chaos, multifractal attractor, nonextensivityComment: Contribution to the proceedings of 2nd International Conference on
News and Expectations in Thermostatistics (NEXT03), Cagliari, Italy,
21-28/09/2003. Submitted to Physica
Nonequilibrium Kinetics of One-Dimensional Bose Gases
We study cold dilute gases made of bosonic atoms, showing that in the
mean-field one-dimensional regime they support stable out-of-equilibrium
states. Starting from the 3D Boltzmann-Vlasov equation with contact
interaction, we derive an effective 1D Landau-Vlasov equation under the
condition of a strong transverse harmonic confinement. We investigate the
existence of out-of-equilibrium states, obtaining stability criteria similar to
those of classical plasmas.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Statistical Mechanics: Theory and Experimen
Fluctuating dynamics at the quasiperiodic onset of chaos, Tsallis q-statistics and Mori's q-phase thermodynamics
We analyze the fluctuating dynamics at the golden-mean transition to chaos in
the critical circle map and find that trajectories within the critical
attractor consist of infinite sets of power laws mixed together. We elucidate
this structure assisted by known renormalization group (RG) results. Next we
proceed to weigh the new findings against Tsallis' entropic and Mori's
thermodynamic theoretical schemes and observe behavior to a large extent richer
than previously reported. We find that the sensitivity to initial conditions
has the form of families of intertwined q-exponentials, of which we determine
the q-indexes and the generalized Lyapunov coefficient spectra. Further, the
dynamics within the critical attractor is found to consist of not one but a
collection of Mori's q-phase transitions with a hierarchical structure. The
value of Mori's `thermodynamic field' variable q at each transition corresponds
to the same special value for the entropic index q. We discuss the relationship
between the two formalisms and indicate the usefulness of the methods involved
to determine the universal trajectory scaling function and/or the ocurrence and
characterization of dynamical phase transitions.Comment: Resubmitted to Physical Review E. The title has been changed slightly
and the abstract has been extended. There is a new subsection following the
conclusions that explains the role and usefulness of the q-statistics in the
problem studied. Other minor changes througout the tex
Intermittency at critical transitions and aging dynamics at edge of chaos
We recall that, at both the intermittency transitions and at the Feigenbaum
attractor in unimodal maps of non-linearity of order , the dynamics
rigorously obeys the Tsallis statistics. We account for the -indices and the
generalized Lyapunov coefficients that characterize the
universality classes of the pitchfork and tangent bifurcations. We identify the
Mori singularities in the Lyapunov spectrum at the edge of chaos with the
appearance of a special value for the entropic index . The physical area of
the Tsallis statistics is further probed by considering the dynamics near
criticality and glass formation in thermal systems. In both cases a close
connection is made with states in unimodal maps with vanishing Lyapunov
coefficients.Comment: Proceedings of: STATPHYS 2004 - 22nd IUPAP International Conference
on Statistical Physics, National Science Seminar Complex, Indian Institute of
Science, Bangalore, 4-9 July 2004. Pramana, in pres
Parallels between the dynamics at the noise-perturbed onset of chaos in logistic maps and the dynamics of glass formation
We develop the characterization of the dynamics at the noise-perturbed edge
of chaos in logistic maps in terms of the quantities normally used to describe
glassy properties in structural glass formers. Following the recognition [Phys.
Lett. \textbf{A 328}, 467 (2004)] that the dynamics at this critical attractor
exhibits analogies with that observed in thermal systems close to
vitrification, we determine the modifications that take place with decreasing
noise amplitude in ensemble and time averaged correlations and in diffusivity.
We corroborate explicitly the occurrence of two-step relaxation, aging with its
characteristic scaling property, and subdiffusion and arrest for this system.
We also discuss features that appear to be specific of the map.Comment: Revised version with substantial improvements. Revtex, 8 pages, 11
figure
Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points
Recently, in Phys. Rev. Lett. 95, 140601 (2005), P. Grassberger addresses the
interesting issue of the applicability of q-statistics to the renowned
Feigenbaum attractor. He concludes there is no genuine connection between the
dynamics at the critical attractor and the generalized statistics and argues
against its usefulness and correctness. Yet, several points are not in line
with our current knowledge, nor are his interpretations. We refer here only to
the dynamics on the attractor to point out that a correct reading of recent
developments invalidates his basic claim.Comment: To be published in Physica
Hamiltonian dynamics reveals the existence of quasi-stationary states for long-range systems in contact with a reservoir
We introduce a Hamiltonian dynamics for the description of long-range
interacting systems in contact with a thermal bath (i.e., in the canonical
ensemble). The dynamics confirms statistical mechanics equilibrium predictions
for the Hamiltonian Mean Field model and the equilibrium ensemble equivalence.
We find that long-lasting quasi-stationary states persist in presence of the
interaction with the environment. Our results indicate that quasi-stationary
states are indeed reproducible in real physical experiments.Comment: Title changed, throughout revision of the tex
Langevin equation in systems with also negative temperatures
We discuss how to derive a Langevin equation (LE) in non standard systems,
i.e. when the kinetic part of the Hamiltonian is not the usual quadratic
function. This generalization allows to consider also cases with negative
absolute temperature. We first give some phenomenological arguments suggesting
the shape of the viscous drift, replacing the usual linear viscous damping, and
its relation with the diffusion coefficient modulating the white noise term. As
a second step, we implement a procedure to reconstruct the drift and the
diffusion term of the LE from the time-series of the momentum of a heavy
particle embedded in a large Hamiltonian system. The results of our
reconstruction are in good agreement with the phenomenological arguments.
Applying the method to systems with negative temperature, we can observe that
also in this case there is a suitable Langevin equation, obtained with a
precise protocol, able to reproduce in a proper way the statistical features of
the slow variables. In other words, even in this context, systems with negative
temperature do not show any pathology.Comment: 20 pages, 7 figures, in press in J. Stat. Mec
Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: rigorous nonextensive solutions
Using the Feigenbaum renormalization group (RG) transformation we work out
exactly the dynamics and the sensitivity to initial conditions for unimodal
maps of nonlinearity at both their pitchfork and tangent
bifurcations. These functions have the form of -exponentials as proposed in
Tsallis' generalization of statistical mechanics. We determine the -indices
that characterize these universality classes and perform for the first time the
calculation of the -generalized Lyapunov coefficient . The
pitchfork and the left-hand side of the tangent bifurcations display weak
insensitivity to initial conditions, while the right-hand side of the tangent
bifurcations presents a `super-strong' (faster than exponential) sensitivity to
initial conditions. We corroborate our analytical results with {\em a priori}
numerical calculations.Comment: latex, 4 figures. Updated references and some general presentation
improvements. To appear published in Europhysics Letter
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