Burr, Lvy, Tsallis

The purpose of this short paper dedicated to the 60th anniversary of Prof.Constantin Tsallis is to show how the use of mathematical tools and physical concepts introduced by Burr, L˙ evy 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. 1 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 uncertainty. In usual thermodynamics, the temperature is a macroscopic observable and the distribution functions are exponentials. Quite generally, one maximizes the Shannon-Boltzmann (S-B) entropy: S = − b a f (x) ln f (x)dx (1)