Burr, Lévy, Tsallis

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Abstract

The purpose of this short paper dedicated to Prof Constantin Tsallis on his 60th anniversary is to show how the use of mathematical tools and physical concepts introduced by Burr, Lévy and Tsallis open a new line of analysis of the old problem of non-Debye decay and universality of relaxation. We also show how a finite characteristic time scale can be expressed in terms of a q-expectation using the concept of q-escort probability. The comparison with the Weron et al. probabilistic theory of relaxation leads to a better understanding of the stochastic properties underlying the Tsallis entropy concept.

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 entropySq=−∫0∞fqq(x)lnqfq(x)dxsubject to the conditions∫0∞fq(x)dx=1,∫0∞gq,i(x)f̃q(x)dx=〈gq,i(x)〉q,i=1,2,…,f̃q(x)=fqq(x)∫abfqq(x)dx,fq(x)=f̃q1/q(x)∫abf̃q1/q(x)dx.The function f̃q(x) 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 Θ̃ as defined by Weron et al. [6]. We have:FΘ(t)=Pr(θ̃<t)=∫0tf(s)ds=1−1+q−12−qtα〈θ̃α〉q−(2−q)/(q−1).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:φα,q

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

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