The Boltzmann-Gibbs formulation of equilibrium statistical mechanics depends crucially on the nature of the Hamiltonian of the JV-body system under study, but this fact is clearly stated only in the introductions of textbooks and, in general, it is very soon neglected. In particular, the very same basic postulate of equilibrium statistical mechanics, the famous Boltzmann principle S = k log W of the microcanonical ensemble, assumes that dynamics can be automatically an easily taken into account, although this is not always justified, as Einstein himself realized [20]. On the other hand, the Boltzmann-Gibbs canonical ensemble is valid only for sufficiently short-range interactions and does not necessarily apply, for example, to gravitational or unscreened Colombian fields for which the usually assumed entropy extensivity postulate is not valid [5]. In 1988, Constantino Tsallis proposed a generalized thermostatistics formalism based on a nonextensive entropic form [24]. Since then, this new theory has been encountering an increasing number of successful applications in different fields (for some recent examples see Abe and Suzuki [1], Baldovin and Robledo [4], Beck et al. [8], Kaniadakis et al. [12], Latora et al. [16], and Tsallis et al. [25]) and seems to be the best candidate for a generalized thermodynamic formalism which should be valid when nonextensivity, long-range correlations, and fractal structures in phase space cannot be neglected: in other words, when the dynamics play a nontrivial role [11] and fluctuations are quite large and non-Gaussian [6, 7, 8, 24, 26]. In this contribution we consider a nonextensive JV-body classical Hamiltonian system, with infinite range interaction, the so-called Hamiltonian mean field (HMF) model, which has been intensively studied in the last several years [3, 13, 14, 15, 17, 18, 19]. The out-of-equilibrium dynamics of the model exhibits a series of anomalies like negative specific heat, metastable states, vanishing Lyapunov exponents, and non-Gaussian velocity distributions. After a brief overview of these anomalies, we show how they can be interpreted in terms of nonextensive thermodynamics according to the present understanding.
PDE/Statistical Mechanics Duality: Relation Between Guerra’s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy
