133 research outputs found
The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
As well known, Boltzmann-Gibbs statistics is the correct way of
thermostatistically approaching ergodic systems. On the other hand, nontrivial
ergodicity breakdown and strong correlations typically drag the system into
out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide
class of such systems, it has been shown in recent years that the correct
approach is to use Tsallis statistics instead. Here we show how the dynamics of
the paradigmatic conservative (area-preserving) standard map exhibits, in an
exceptionally clear manner, the crossing from one statistics to the other. Our
results unambiguously illustrate the domains of validity of both
Boltzmann-Gibbs and Tsallis statistics
Nonextensive statistical mechanics - Applications to nuclear and high energy physics
A variety of phenomena in nuclear and high energy physics seemingly do not
satisfy the basic hypothesis for possible stationary states to be of the type
covered by Boltzmann-Gibbs (BG) statistical mechanics. More specifically, the
system appears to relax, along time, on macroscopic states which violate the
ergodic assumption. Some of these phenomena appear to follow, instead, the
prescriptions of nonextensive statistical mechanics. In the same manner that
the BG formalism is based on the entropy , the
nonextensive one is based on the form (with
). Typically, the systems following the rules derived from the
former exhibit an {\it exponential} relaxation with time toward a stationary
state characterized by an {\it exponential} dependence on the energy ({\it
thermal equilibrium}), whereas those following the rules derived from the
latter are characterized by (asymptotic) {\it power-laws} (both the typical
time dependences, and the energy distribution at the stationary state). A brief
review of this theory is given here, as well as of some of its applications,
such as electron-positron annihilation producing hadronic jets, collisions
involving heavy nuclei, the solar neutrino problem, anomalous diffusion of a
quark in a quark-gluon plasma, and flux of cosmic rays on Earth. In addition to
these points, very recent developments generalizing nonextensive statistical
mechanics itself are mentioned.Comment: 23 pages including 5 figures. To appear in the Proceedings of the Xth
International Workshop on Multiparticle Production - Correlations and
Fluctuations in QCD (8-15 June 2002, Crete), ed. N. Antoniou (World
Scientific, Singapore, 2003). It includes a reply to the criticism expressed
in R. Luzzi, A.R. Vasconcellos and J.G. Ramos, Science 298, 1171 (2002
Time evolution of nonadditive entropies: The logistic map
Due to the second principle of thermodynamics, the time dependence of entropy
for all kinds of systems under all kinds of physical circumstances always
thrives interest. The logistic map
is neither large, since it has only one degree of freedom, nor closed, since it
is dissipative. It exhibits, nevertheless, a peculiar time evolution of its
natural entropy, which is the additive Boltzmann-Gibbs-Shannon one,
, for all values of for which the
Lyapunov exponent is positive, and the nonadditive one with at the edge of chaos,
where the Lyapunov exponent vanishes, being the number of windows of the
phase space partition. We numerically show that, for increasing time, the
phase-space-averaged entropy overshoots above its stationary-state value in all
cases. However, when , the overshooting gradually disappears for
the most chaotic case (), whereas, in remarkable contrast, it appears to
monotonically diverge at the Feigenbaum point (). Consequently,
the stationary-state entropy value is achieved from {\it above}, instead of
from {\it below}, as it could have been a priori expected. These results raise
the question whether the usual requirements -- large, closed, and for generic
initial conditions -- for the second principle validity might be necessary but
not sufficient.Comment: 7 pages, 6 composed figures (total of 15 simple figures
- β¦