Physical Origin of the Power-Law Tailed Statistical Distributions

Abstract

Starting from the BBGKY hierarchy, describing the kinetics of nonlinear particle system, we obtain the relevant entropy and stationary distribution function. Subsequently, by employing the Lorentz transformations we propose the relativistic generalization of the exponential and logarithmic functions. The related particle distribution and entropy represents the relativistic extension of the classical Maxwell-Boltzmann distribution and of the Boltzmann entropy respectively and define the statistical mechanics presented in [Phys. Rev. E 66, 056125 (2002)] and [Phys. Rev. E 72, 036108 (2005). The achievements of the present effort, support the idea that the experimentally observed power law tailed statistical distributions in plasma physics, are enforced by the relativistic microscopic particle dynamics.

Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions

“…The above κ-distribution at low energies is the ordinary Boltzmann distribution while at high energies presents an power-law tail. For this reason the statistical theory [1][2][3][4][5][6][7][8][9][10], based on the distribution (1.1) has attracted the interest of many researchers. In the last twelve years various authors have considered the foundations of the statistical theory based on the κdistribution, in connection with the historical evolution of the research on the power-law tailed statistical distributions [7,11] e.g.…”

“…The above κ-distribution at low energies is the ordinary Boltzmann distribution while at high energies presents an power-law tail. For this reason the statistical theory [1][2][3][4][5][6][7][8][9][10], based on the distribution (1.1) has attracted the interest of many researchers.…”

Kaniadakis Entropy Leads to Particle–Hole Symmetric Distribution

“…There are several generailzed entropy formulas and corresponding canonical distributions [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ]. At their core, they can be viewed as the generalization of the logarithm and exponential functions while keeping their inverse roles.…”

“…The relation to special relativity is discussed in [ 9 ], and to the Boltzmann equation in [ 10 ]. Fractional statistics in kappa statistics are dealt with in [ 11 ], nonlinear kinetics in [ 12 ], and a general review about the physical origins in [ 13 ].…”

Relativistic Roots of κ-Entropy

“…Remarkably, although the statistical theory based on S κ can be traced back to the first principles of special relativity, it can also be introduced without reference to special relativity, as will be shown in Section II, since it also has applications outside relativistic physics. For this reason, statistical theory [7][8][9][10][11][12][13] based on the κ-distribution has attracted the interest of many researchers. In the last two decades, various authors have devoted themselves to the study of both the theoretical foundations of the theory and its applications not only in plasma physics but also in various other areas of the science of complex physical, natural, or artificial statistical systems.…”