Two generalizations of the Boltzmann equation

Abstract

We connect two different extensions of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint part. Direct transformation formulas between key functions of the two approaches are given.

Fluctuations, correlations and the nonextensivity

“…However, in any possible applications one has to remember that eq. (12) has been obtained only for correlations caused by fluctuations and therefore cannot be used for estimations of the role of correlations as such in Tsallis dis-tributions. In Appendix A we present simple example of correlations making Tsallis entropy additive (cf.…”

Power laws in elementary and heavy-ion collisions

“…For example, in [29] it was a random distortion of energy and momentum conservation caused by the surrounding system which resulted in the emergence of some nonextensive equilibrium. In [30,31] the two-body energy composition in transport theory formulation of the collision process is replaced by a generalized energy sum, h(E 1 , E 2 ), which is assumed to be associative but which is not necessarily simple addition and contains contributions stemming from pair interaction (in the simplest case). It turns out that under quite general assumptions about the function h, a division of the total energy among free particles is possible.…”

Nonextensive perfect hydrodynamics — a model of dissipative relativistic hydrodynamics?

“…This results in the emergence of some nonextensive equilibrium. In [55][56][57][58][59] the two-body energy composition is replaced by some generalized energy sum (E 1 E 2 ), which is assumed to be associative but which is not necessarily simple addition and contains contributions stemming from pair interaction (in the simplest case). It turns out that, under quite general assumptions about the function , the division of the total energy among free particles can be done.…”

“…By postulating Eq. (1) we are, in fact, assuming that, instead of a strict (local) equilibrium, a kind of stationary state is being formed, which already includes some interactions and which is summarily characterized by a parameter ; very much in the spirit of [54][55][56][57][58][59][60] mentioned before. With this assumption we are already departing from the picture of the usual ideal fluid with its local thermal equilibrium, which is the prerogative of ideal hydrodynamics [64,65].…”