**70**, 3-13 (2009)

https://doi.org/10.1140/epjb/e2009-00161-0

## Maximum entropy principle and power-law tailed distributions

Dipartimento di
Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Corresponding author: ^{a}
giorgio.kaniadakis@polito.it

Received:
17
December
2008

Published online:
5
May
2009

In ordinary statistical mechanics the Boltzmann-Shannon
entropy is related to the Maxwell-Bolzmann distribution
p_{i} by means of a twofold link. The first link is differential and
is offered by the Jaynes Maximum Entropy Principle. Indeed, the
Maxwell-Boltzmann distribution is obtained by maximizing the Boltzmann-Shannon entropy under
proper constraints. The second link is algebraic and imposes that
both the entropy and the distribution must be expressed in
terms of the same function in direct and inverse form. Indeed, the
Maxwell-Boltzmann distribution p_{i} is expressed in terms of the exponential
function, while the Boltzmann-Shannon entropy is defined as the mean value of
-ln (p_{i}).
In generalized statistical mechanics the second link is customarily
relaxed. Of course, the generalized exponential function defining
the probability distribution function after inversion, produces a
generalized logarithm Λ(p_{i}). But, in general, the mean value
of -Λ(p_{i}) is not the entropy of the system. Here we
reconsider the question first posed in [Phys. Rev. E **66**,
056125 (2002) and **72**, 036108 (2005)], if and how is it
possible to select generalized statistical theories in which the
above mentioned twofold link between entropy and the distribution
function continues to hold, such as in the case of ordinary
statistical mechanics.
Within this scenario, apart from the standard
logarithmic-exponential functions that define ordinary statistical
mechanics, there emerge other new couples of direct-inverse
functions, i.e. generalized logarithms Λ(x) and generalized
exponentials Λ^{-1}(x), defining coherent and
self-consistent generalized statistical theories. Interestingly, all
these theories preserve the main features of ordinary statistical
mechanics, and predict distribution functions presenting power-law
tails. Furthermore, the obtained generalized entropies are both
thermodynamically and Lesche stable.

PACS: 05.20.-y – Classical statistical mechanics / 51.10.+y – Kinetic and transport theory of gases / 03.30.+p – Special relativity / 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009*