Conservative force fields in non-Gaussian statistics

Abstract

In this letter, we determine the κ-distribution function for a gas in the presence of an external field of force described by a potential U(r). In the case of a dilute gas, we show that the κ-power law distribution including the potential energy factor term can rigorously be deduced in the framework of kinetic theory with basis on the Vlasov equation. Such a result is significant as a preliminary to the discussion on the role of long range interactions in the Kaniadakis thermostatistics and the underlying kinetic theory.

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

“…On the other hand specific applications to physical systems have been considered, e.g. the cosmic rays [3], relativistic [37] and classical [38] plasmas in presence of external electromagnetic fields, the relaxation in relativistic plasmas under wave-particle interactions [39,40], anomalous diffusion [41,42], nonlinear kinetics [43][44][45], kinetics of interacting atoms and photons [46], particle kinetics in the presence of temperature gradients [47,48], particle systems in external conservative force fields [49], stellar distributions in astrophysics [50][51][52][53], quark-gluon plasma formation [54], quantum hadrodynamics models [55], the fracture propagation [56], etc. Other applications concern dynamical systems at the edge of chaos [57][58][59], fractal systems [60], field theories [61], the random matrix theory [62][63][64], the error theory [65], the game theory [66], the theory of complex networks [67], the information theory [68], etc.…”

Relativistic entropy and related Boltzmann kinetics

“…On the other hand, specific applications of the theory, include e.g. the cosmic rays [18], relativistic [31] and classical [32] plasmas in presence of external electromagnetic fields, the relaxation in relativistic plasmas under wave-particle interactions [33,34], astrophysical systems [35,36], the kinetics of interacting atoms and photons [37], particle systems in external conservative force fields [38], the quark-gluon plasma formation [39] etc. Other applications regard dynamical systems at the edge of chaos [40,41], fractal systems [42], the random matrix theory [43], the error theory [44], the game theory [45], the Information theory [46], etc.…”

Physical Origin of the Power-Law Tailed Statistical Distributions

“…It is worth stressing that the latter relationship defines the relativistic entropy, as proportional to the mean value of − ln κ (f /ǫ), like in the case of classical statistical mechanics where the Boltzmann entropy, S = − d 3 p f ln (f /e), is proportional to the mean value of − ln(f /e). Clearly the entropy (42), in the classical limit reduces to the Boltzmann entropy. The constant γ, given by (40), represents the Lorentz factor related to the velocity v * appearing in (25), and in the classical limit, approaches the unity.…”

“…On the other hand specific applications to physical systems have been considered, e.g. the cosmic rays [4], relativistic [32] and classical [33] plasmas in presence of external electromagnetic fields, the relaxation in relativistic plasmas under wave-particle interactions [34,35], anomalous diffusion [36,37], non-linear kinetics [38,39], kinetics of interacting atoms and photons [40], particle kinetics in the presence of temperature gradients [41], particle systems in external conservative force fields [42], stellar distributions in astrophysics [43][44][45], quark-gluon plasma formation [46], quantum hadrodynamics models [47], the fracture propagation [48], etc. Other applications regard dynamical systems at the edge of chaos [49][50][51], fractal systems [52], field theories [53], the random matrix theory [54,55], the error theory [56], the game theory [57], the information theory [58], etc.…”