Non-Extensive Statistical Mechanics

Standard statistical mechanics is extensive, i.e. the Boltzmann-Gibbs (BG) entropy of two subsets of a gas is just the sum of the two because of the short-range nature of the interaction. But many objects in nature interact through long-range interactions, think of gravitational or unscreened Coulomb forces. Therefore the property of additivity (extensivity) is very often violated. This is simply forgotten in standards thermostatistics books apart from some sentences of warning in the very first pages. In order to take this fact into account, Constantino Tsallis in 1988, proposed a new formula for the entropy which contains the standard Boltzmann-Gibbs one as a limiting case. Originally this generalized entropy resulted to be non-extensive in absence of correlations, therefore the name "non-extensive statistical mechanics", but recently has been shown that it becomes extensive for strongly correlated systems (for these systems is the BG entropy which is non-extensive!). In the last decade the Tsallis' entropy has found many applications in plasma physics, astrophysics, fractals, turbulence, finance, systems with long-range correlations and complex systems. In practice the new generalized thermostatistics reduces to the standard one when the interaction becomes short-ranged. Moreover, all the known theorems for standard statistics mechanics hold also in the new generalized form.

First among Cactus Group's people, prof. A.Rapisarda got interested in the generalized non extensive statistical mechanics after a meeting at MIT in the fall of 1998 where Tsallis presented his new entropy. Then V. Latora, M. Baranger (MIT), Constantino Tsallis (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro) and Rapisarda himself applied these concepts to understand the growth of the generalized entropy in the case of the logistic map at the edge-of-chaos, i.e. just at the critical point when the behavior of the logistic map changes from regular to chaotic. Their first paper on this subject was published in Physics Letters A (2000) and a second one is Chaos Solitons and Fractals. More recently also A.Pluchino was involved in this field and a great part of his PhD thesis was dedicated to the non-extensive interpretation of metastability in the Hamiltonian Mean Field model. See below for other papers on generalized thermostatistics published in several journals.