Abstract composition rule for relativistic kinetic energy in the thermodynamical limit

Abstract

We demonstrate by simple mathematical considerations that a power-law-tailed distribution in the kinetic energy of relativistic particles can be a limiting distribution seen in relativistic heavy-ion experiments. We prove that the infinite repetition of an arbitrary composition rule on an infinitesimal amount leads to a rule with a formal logarithm. As a consequence the stationary distribution of energy in the thermodynamical limit follows the composed function of the Boltzmann-Gibbs exponential with this formal logarithm. In particular, interactions described as solely functions of the relative four-momentum squared lead to kinetic energy distributions of the Tsallis-Pareto (cut power law) form in the high-energy limit.

Non-extensive equilibration in relativistic matter

“…That such rules necessarily arise in the thermodynamical limit is demonstrated in Ref. [20]. Using formal logarithms all the classical concepts and techniques can be applied to describe thermal equilibrium or to generate distributions accordingly.…”

“…The infinite repetition of an arbitrary pairwise, iterable composition rule is an associative rule [20]. It is a mathematical property that associative rules always possess a strict monotonic function, called here the formal logarithm, in terms of which they can be expressed [30].…”

“…There are several speculations on possible causes for such deformations, measured in a summerized way by a parameter used in Tsallis' non-extensive [8][9][10][11][12] entropy formula, or Rényi's extensive one [13]. Fluctuations of temperature or energy equipartition [14][15][16], anomalous diffusion [17] or multiplicative noise [18,19], or alternatively an altering in the two-body energy composition rule [20][21][22][23][24] were investigated in theoretical approaches. In order to solidify the motivation for such an approach we sketch some arguments about possible occurences of non-extensivity in thermodynamically treateble, large systems (the size is measured not by a volume, but by the number of particles produced and analyzed).…”

Non-Extensive Entropic Distance Based on Diffusion: Restrictions on Parameters in Entropy Formulae

“…Another way to obtain a generalized entropy formula starts with physical properties, like the universality of the thermal equilibrium between two systems as described by the zeroth law of thermodynamics [12,13], and by this it also includes the associativity assumption [14]. It is also noteworthy that starting with a general, non-associative composition prescription one arrives asymptotically at an associative one-only by repeating it in small steps and reconstructing the effective composition formula in the continuous scaling limit [15].…”

Lattice gauge theory with fluctuating temperature

“…Fits using the Tsallis distribution to the observed particle spectra in heavy-ion reactions have also been remarkably successful [17,18]. Non-conventional distributions are based in general on nonadditive composition rules [19], or on a non-conventional entropy formula (which replaces the Boltzmann entropy). Such an entropy formula is the Tsallis entropy [15], which is not an extensive quantity.…”