Statistical mechanics in the context of special relativity. II.

Abstract

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

Nonextensivity and the q-distribution of a relativistic gas under an external electromagnetic field

“…For the above relativistic gas in the presence of an external electromagnetic field, it can be written [26,27] as…”

“…The states of this gas can be characterized by a Lorentz invariant one-particle distribution function, f q (x, p). Thus at each time t, f q (x, p)d 3 xd 3 p gives the number of particles in the volume element d 3 xd 3 p around the space position x and momentum p. The evolution equation of the relativistic distribution function is assumed to be the relativistic generalized Boltzmann equation in the qkinetic theory [26]: …”

“…The generalized Boltzmann equation and the q-H theorem for a relativistic case are studied in the q-kinetic theory, where an expression of the nonextensive four-entropy flux is given [26] by…”

A method to deconvolve stellar rotational velocities

“…Kaniadakis (2002Kaniadakis ( , 2005, based on the same concept of maximum entropy, developed a power-law statistics, where the distribution function is given by:…”

“…They obtained a much better fit when the assumption of Gibbs entropy in standard statistical mechanics is released and distribution functions from nonextensive statistical mechanics are applied to the sample. Specifically, they used the Tsallis distribution (Tsallis 1998) and the Kaniadakis power-law distribution (Kaniadakis 2002(Kaniadakis , 2005, both based on the concept of maximum entropy (Gell-Mann & Tsallis 2004). These distributions are also plotted in Fig.…”