Nonadditive entropy: the concept and its use. European Phys. J. A 40, 257-266

Santa Fe Institute 1399 Hyde Park Road 87501 Santa Fe USA
European Physical Journal A (Impact Factor: 2.74). 12/2008; 40(3):257-266. DOI: 10.1140/epja/i2009-10799-0
Source: arXiv


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 ∑Wi=1p
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.

Full-text preview

Available from: ArXiv
  • Source
    • "black holes – along the same lines of reasoning. We believe that this can either be achieved by relaxing the requirement of full thermodynamic equilibrium or by using different statistics, such as Rényi or Tsallis' statistics [39] [40] "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this work we prove that the maximally symmetric vacuum solutions of General Relativity emerge from the geometric structure of statistical mechanics and thermodynamic fluctuation theory. To present our argument, we begin by showing that the pseudo-Riemannian structure of the Thermodynamic Phase Space is a solution to the vacuum Einstein-Gauss-Bonnet theory of gravity with a cosmological constant. Then, we use the geometry of equilibrium thermodynamics to demonstrate that the maximally symmetric vacuum solutions of Einstein's Field Equations -- Minkowski, de-Sitter and Anti-de-Sitter spacetimes -- correspond to thermodynamic fluctuations. Moreover, we argue that these might be the only possible solutions that can be derived in this manner. Thus, the results presented here are the first concrete examples of spacetimes effectively emerging from the thermodynamic limit over an unspecified microscopic theory without any further assumptions.
    Full-text · Article · Mar 2015
  • Source
    • "Indeed as Bouchaud and Potter point out, the equation (7) is linearly related to the Tsallis [8] [9] entropy function and the entire process of obtaining the weights, p i is equivalent to minimizing a free -utility function: "
    [Show abstract] [Hide abstract]
    ABSTRACT: We briefly review the approach to optimization of portfolios according to the theory of Markowitz and propose a further modification that can improve the outcome of the optimization process. The modification takes account of the entropic contribution from the time series used to compute the parameters in the Markowitz method.
    Full-text · Article · Aug 2014
  • Source
    • "During the last years some papers have been published in which again Gibbs Paradox (non-additive entropy), the concept of entropy and the mixing process of particle systems were considered, see for example [1] [2] [3] [4] [5] [6]. From the definition of additivity (or non-additivity) follows that this notion is related to a system composed of several components, S=∑ i S i i≥2. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this work the non-additive entropy is examined. It appears in isolated particle systems composed of few components. Therefore, the mixing of isolated particle systems S=S1+S2 has been studied. Two cases are considered T1=T2 and T1\leqT2, where T1,T2 are the initial temperatures of the system S1 and S2 respectively. The concept of similar systems containing interacting particles is introduced. These systems are defined by a common temperature and an identical time evolution process, i.e. the approach to the same thermodynamic equilibrium. The main results are: 1) The properties of the similar particle systems yield the non-additive entropy and free energy. The Gibbs Paradox is not a paradox. 2) The relation between the initial temperatures T1 and T2 governs the mixing process. 3) In the two cases T1=T2, T1\leqT2 mixing of the systems S1, S2 results in a uniform union system S=S1+S2. The systems S, S1, S2 are similar one to the other. 4) The mixing process is independent of the extensive quantities (volume, particle number, energy) and of the particle type. Only the mean energy plays an important role in the mixing of the systems S1, S2. 5) Mixing in the case T1\leqT2 is in essence a thermalization process, but mixing in the case T1=T2 is not a thermodynamic process. 6)Mixing is an irreversible process. Keywords: Entropy; Similar systems of interacting particles; Mixing of systems; Thermal equilibrium
    Preview · Article · Jun 2012
Show more