Physica A: Statistical Mechanics and its Applications
Burr, Lévy, Tsallis
Section snippets
Maximum entropy principle and probability distributions
Most of the probability distributions used in natural, biological, social and economic sciences can be formally derived by maximizing the entropy with adequate constraints (maxS principle) [1]. According to the maxS principle, given some partial information about a random variable, i.e., the knowledge of related macroscopic measurable quantities (macroscopic observables), one should choose for it, the probability distribution that is consistent with that information, but has otherwise a maximum
Maximization with Tsallis entropy
A generalization of the S–B entropy is appropriate when the phenomena are described by distributions with a Lévy characteristic index αL<1. Here, we will start from the Tsallis non-extensive entropysubject to the conditionsThe function is the so-called “escort” probability [3], [4]. To generalize what has been done with the standard S–B entropy, we can consider
Universality in non-Debye relaxation
This result can be used to represent non-Debye relaxation if we identify the random variable X with the macroscopic waiting time as defined by Weron et al. [6]. We have:The relaxation function φ(t) can be written as the survival probability of the non-equilibrium initial state of the relaxing system. Its value is determined by the probability that the system as a whole will not make transition out of its original state until time t:
Stochastic theory and Tsallis entropy
The discussion of the equivalence of the two results obtained for the “universal” relaxation function (22) and (23) from the maxSq principle and the Weron et al. stochastic theory [6], can lead to a better understanding of the stochastic properties underlying the Tsallis entropy concept as well as its possible generalization for more complex spatio-temporal dynamical systems.
The starting point is the observation of the fact that the relaxation function for the whole system, i.e., its survival
Conclusions
In the case of non-Debye relaxation, the short-range interactions, geometric and dynamic correlations can be accounted for by maximizing the S–B entropy with adequate constraints. The small clusters and short-time relaxation can be described by a stretched exponential relaxation function. The exponent α⩽1 introduced in the constraints is a measure of the “non-idealness” of the relaxation processes at this scale (fluctuations of the size and intra-cluster interactions). The system is extensive
References (13)
- et al.
Physica A
(1998) Maximum Entropy Models in Science and Engineering
(1989)- et al.
Non-Debye Relaxation in Condensed Matter
(1987) Biometrika
(1977)- et al.
J. Stat. Phys
(1997)