Nonadditive entropy: The concept and its use

Abstract

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential dS = \( \delta\) Q/T , where \( \delta\) 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) \(\ensuremath S_{BG}=-k\sum_{i=1}^W p_i \ln p_i\) , 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., \(\ensuremath p_{ij}^{A+B}=p_i^A p_j^B, \forall(i,j)\) , we verify that \(\ensuremath S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)\) . 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 SBG to be extensive in the thermodynamical sense, i.e., that \(\ensuremath S_{BG}(N) \propto N\) in the N ≫ 1 limit. If, on the contrary, the correlations between the N elements are strong enough, then the extensivity of SBG 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 \(\ensuremath S_q=k [1-\sum_{i=1}^W p_i^q]/(q-1) (q\in{R}; S_1=S_{BG})\) . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form Sq is, for any q \( \neq\) 1 , nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies \(\ensuremath S_q(A+B)/k=[S_q(A)/k]+ [S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k] \neq S_q(A)/k+S_q(B)/k\) . 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 qent (where ent stands for entropy . In other words, for such systems, we verify that \(\ensuremath S_{q_{ent}}(N) \propto N (N \gg 1)\) , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which SBG 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 Sq is extensive. Such systems are out of the present scope: they might need forms of entropy different from Sq, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with Sq, 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.