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 $S_{BG}=-k \sum_i p_i \ln p_i$, the nonextensive one is based on the form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$). 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 $x_{t+1}=1-a x_t^2 \in [-1,1]\;(a\in [0,2])$ 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, $S_{BG}=-\sum_{i=1}^W p_i \ln p_i$, for all values of $a$ for which the Lyapunov exponent is positive, and the nonadditive one $S_q= \frac{1-\sum_{i=1}^W p_i^q}{q-1}$ with $q=0.2445\dots$ at the edge of chaos, where the Lyapunov exponent vanishes, $W$ 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 $W\to\infty$, the overshooting gradually disappears for the most chaotic case ($a=2$), whereas, in remarkable contrast, it appears to monotonically diverge at the Feigenbaum point ($a=1.4011\dots$). 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