**40**: 257-266

https://doi.org/10.1140/epja/i2009-10799-0

Regular Article - Theoretical Physics

## Nonadditive entropy: The concept and its use

^{1}
Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Xavier Sigaud 150, 22290-180, Rio de Janeiro-RJ, Brazil

^{2}
Santa Fe Institute, 1399 Hyde Park Road, 87501, Santa Fe, USA

^{*} e-mail: tsallis@cbpf.br

Received:
31
January
2009

Accepted:
15
February
2009

Published online:
20
May
2009

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential *dS* =
*Q*/*T* , where
*Q* is the heat transfer and the absolute temperature *T* its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this quantity can be expressed in terms of the probabilities associated with the microscopic configurations of the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in its discrete form) , where *k* is the Boltzmann constant, and {*p*
_{i}} the probabilities corresponding to the *W* microscopic configurations (hence ∑^{W}
_{i=1}
*p*
_{i} = 1 . This entropic form, further discussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical mechanics, is *additive*. Indeed, if we consider a system composed by any two probabilistically independent subsystems *A* and *B* (*i.e.*, , we verify that . If a system is constituted by *N* equal elements which are either independent or quasi-independent (*i.e.*, not too strongly correlated, in some specific *nonlocal* sense), this additivity guarantees S_{BG} to be *extensive* in the thermodynamical sense, *i.e.*, that in the *N* ≫ 1 limit. If, on the contrary, the correlations between the *N* elements are strong enough, then the extensivity of S_{BG} is lost, being therefore incompatible with classical thermodynamics. In such a case, the many and precious relations described in textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty, and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the entropy . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form S_{q} is, for any *q*
1 , *nonadditive*. Indeed, for two probabilistically independent subsystems, it satisfies . This form will turn out to be *extensive* for an important class of nonlocal correlations, if *q* is set equal to a special value different from unity, noted q_{ent} (where *ent* stands for *entropy* . In other words, for such systems, we verify that , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which S_{BG} is extensive, obviously correspond to *q*
_{ent} = 1 . Quite complex systems exist in the sense that, for them, no value of *q* exists such that S_{q} is extensive. Such systems are out of the present scope: they might need forms of entropy different from S_{q}, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with S_{q}, the *q* -generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of *q* -exponentials, *q* -Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations --in high-energy physics and elsewhere-- are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms *versus* distinct regimes of a single physical mechanism.

PACS: 05.20.-y Classical statistical mechanics – / 02.50.Cw Probability theory – / 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems – / 05.70.-a Thermodynamics –

*© SIF, Springer-Verlag Berlin Heidelberg, 2009*