2002/10/31 by G. Kaniadakis · 13 citations
Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Complex Systems and Time Series Analysis #Statistical Mechanics and Entropy #cond-mat.soft #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.66.056125
published as Phys. Rev. E 66, 056125 (2002) · 17 pages (two columns), 5 figures, RevTeX4, minor typing corrections
openalex publication_date 2002/11/25 · arxiv created 2002/11/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In Ref. [Physica A 296, 405 (2001)], starting from the one parameter deformation of the exponential function exp_\ensuremathκ(x)=(√1+\ensuremathκ2x2+\ensuremathκx)^1/\ensuremathκ, a statistical mechanics has been constructed which reduces to the ordinary Boltzmann-Gibbs statistical mechanics as the deformation parameter \ensuremathκ approaches to zero. The distribution f=exp_\ensuremathκ(\ensuremath-\ensuremathβE+\ensuremathβ\ensuremathμ) obtained within this statistical mechanics shows a power law tail and depends on the nonspecified parameter \ensuremathβ, containing all the information about the temperature of the system. On the other hand, the entropic form S_\ensuremathκ=\ensuremath∫d3p(c_\ensuremathκf^1+\ensuremathκ+c_\ensuremath-\ensuremathκf^1\ensuremath-\ensuremathκ), which after maximization produces the distribution f and reduces to the standard Boltzmann-Shannon entropy S0 as \stackrel\ensuremath→\ensuremathκ0, contains the coefficient c_\ensuremathκ whose expression involves, beside the Boltzmann constant, another nonspecified parameter \ensuremathα. In the present effort we show that S_\ensuremathκ is the unique existing entropy obtained by a continuous deformation of S0 and preserving unaltered its fundamental properties of concavity, additivity, and extensivity. These properties of S_\ensuremathκ permit to determine unequivocally the values of the above mentioned parameters \ensuremathβ and \ensuremathα. Subsequently, we explain the origin of the deformation mechanism introduced by \ensuremathκ and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter \ensuremathκ which results to depend on the light speed c and reduces to zero as \stackrel\ensuremath→c\ensuremath∞ recovering in this way the ordinary statistical mechanics and thermodynamics. The statistical mechanics here presented, does not contain free parameters, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic, and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data.