Nonextensive statistical mechanics: Some links with astronomical phenomena

Abstract

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann-Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy $S_{BG}=-k{\sum }_i{p}_i\ln p_i$, the nonextensive one is based on the form $S_q=k(1-{\sum }_i{p}_i^q)/(q-1)$ (with $S_1=S_{BG}$). The stationary states of the former are characterized by an {\it exponential} dependence on the energy, whereas those of the latter are characterized by an (asymptotic) {\it power-law}. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits {\it versus} the Ptolemaic epicycles is developed, where we show that optimizing $S_q$ with a few constraints is equivalent to optimizing $S_{BG}$ with an infinite number of constraints.

23 pages including one EPS figure. To appear in the Proceedings of the XIth United Nations / European Space Agency Workshop on Basic Space Sciences, Office for Outer Space Affairs / United Nations (Cordoba, 9-13 September 2002), ed. H. Haubold, special issue of Astrophysics and Space Science (Kluwer Academic Publishers, Dordrecht, 2003). It includes a reply to the criticism by M. Nauenberg, cond-mat/0210561 [version 2]

References in corpus (4)

• Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
• A critique of non-extensive q-entropy for thermal statistics
• Gravitational instability of isothermal and polytropic spheres
• Anomalous diffusion, nonlinear fractional Fokker-Planck equation and solutions

Cited by in corpus (11)

• Nonextensivity in nonequilibrium plasma systems with Coulombian long-range interactions
• What should a statistical mechanics satisfy to reflect nature?
• Dark matter distribution function from non-extensive statistical mechanics
• On the meaning of Tsallis functional in astrophysics
• What does the nonextensive parameter q stand for in self-gravitating systems ?
• Cluster temperatures and non-extensive thermo-statistics
• The hydrostatic equilibrium and Tsallis equilibrium for self-gravitating systems
• Formal equivalence between Tsallis and extended Boltzmann-Gibbs statistics
• Entropic Upper Bound on Gravitational Binding Energy
• Crab pulsar spectrum: a non-extensive approach
• On a heuristic point of view related to quantum nonequilibrium statistical mechanics