Nonextensive Entropy-Interdisciplinary Applications
Edited by Murray Gell-Mann and Constantino Tsallis
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The present book constitutes a pedagogical effort that reflects the presentations and discussion at the International Workshop on "Interdisciplinary Applications of Ideas from Nonextensive Statistical Mechanics and Thermodynamics," held at the Santa Fe Institute in New Mexico from April 8--12, 2002. The participants, close to 60 in number, were scientists at both junior and senior levels from Argentina, Brazil, Canada, Germany, Great Britain, Italy, Mexico, Poland, and the U.S.A. The subjects of the chapters relate to dynamical, physical, geophysical, biological, economic, financial, and social systems, and to networks, linguistics, and plectics.
Some of the contributions focus directly on a specific nonextensive entropic form proposed by one of us as a basis for generalizing Boltzmann-Gibbs (BG) statistical mechanics to a formalism sometimes called nonextensive statistical mechanics. Such is the case for the chapters by Abe; Andrade, Almeida, Moreira, Adib, and Farias; Baldovin; Borland; Lyra; Montemurro; Osorio, Borland, and Tsallis; Penna, Sartorelli, Pinto, and Gonc{c}alves; Rapisarda and Latora; Robledo; Plastino, Martin, and Rosso; Scafetta, Grigolini, Hamilton, and West; Touchette; Wio; and Tsallis. Other contributions analyze problems that point toward possible fruitful uses of nonextensive concepts. Such is the case for the manuscripts by Cannas, Marco, P'aez, and Montemurro; Gell-Mann and Lloyd; Latora and Marchiori; and Stinchcombe. Finally, other contributions, such as those of Buiatti, Bogani, Acquisti, Mersi, and Fronzoni; Bunde, Eichner, Govindan, Havlin, Koscielny-Bunde, Rybski, and Vjushin; Debowski, andbreak P'erez-Mercader, focus on rich complex systems that might or might not be related to nonextensive entropy.
It should be clear that nonextensive statistical mechanics is by no means intended to replace BG statistical mechanics for systems such as those in stationary states characterized by thermal equilibrium consistent with ergodicity. The nonextensive alternative is proposed, instead, as a way of dealing, through mathematical methods that are quite similar to the usual ones, with anomalous systems. These include a wide class of nonergodic systems, with stationary (or rather quasistationary) states that are metastable and long-lived, for example in many-body Hamiltonian systems with long-range forces. Some strong analogies with the BG theory emerge, as well as some important physical differences. Nonextensive statistical mechanics exhibits apparent success (in a sense that we discuss below) for certain closed systems as well as a variety of open systems---in biology, economics, linguistics, the physics of turbulence, and other fields. An intriguing question that remains unanswered is: exactly what do all these systems have in common? One suspects, of course, that the deep explanation must arise from microscopic dynamics. The various cases could all be associated with something like a scale-free dynamical occupancy of phase space, but this certainly deserves further investigation.
Are other, somewhat similar generalizations of statistical mechanics also possible? What would be the physical entropy to be used, and what would be its relation to the symmetry of occupation of phase space? Answering such questions is by no means trivial. However, it is interesting to remark that for $q$ positive the present nonextensive entropy shares with the BG entropy three important properties, namely concavity (related to thermodynamic stability, or robustness with respect to fluctuations of energy and other quantities), stability or continuity (related to experimental robustness, in the sense that similar experiments should provide quantitatively similar results), and finiteness of the entropy production per unit time (conveniently characterizing the gradual exploration of the available phase space). In order to appreciate the difficulty of satisfying all of these conditions, it is worth mentioning that R'enyi entropy violates all three.
Since the nonextensive entropy Sq and the R'enyi entropy are related through a monotonic function, the distinction between them is irrelevant to the determination of the probability distribution, but for other physical quantities, such as those connected with the flow of entropy, the distinction can be crucial. In general, one can vary the form of the entropy and also the choice of average quantities to be kept fixed as the entropy is maximized, while keeping the probabilities unchanged. Again, quantities that explicitly involve the entropy will come out different. It is important to remember that the probabilities alone do not distinguish in a rigorous manner between conventional and unconventional entropy. However, the probabilities can suffice to distinguish different formulae for entropy if the average quantities kept fixed are restricted to particular ones that are regarded as natural, for example linear or quadratic expressions in the variable under study. It is in this specific sense that nonextensive entropy has scored much of its success.
We are extremely pleased to acknowledge the very valuable support of Ellen Goldberg, the editorial assistance of Ronda Butler-Villa, Della Ulibarri, and Laura Ware, the practical help on so many occasions of Olivia Posner, Kevin Drennan, and Andi Sutherland, and---last but not least---the generous financial support of the Santa Fe Institute, International Program grant, without which the meeting could not have occurred.
Constantino Tsallis, Centro Brasileiro de Pesquisas Fisicas
Murray Gell-Mann, Santa Fe Institute