On the Simplification of Statistical Mechanics for Space Plasmas

George Livadiotis

doi:10.3390/e19060285

Abstract

Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin of kappa distributions, and shows that they can be connected with non-extensive statistical mechanics without considering the dyadic formalism of ordinary/escort distributions. While this concept does significantly simplify the usage of the theory, it costs the definition of a dyadic entropic formulation, in order to preserve the consistency between statistical mechanics and thermodynamics. Therefore, the simplification of the theory by means of avoiding dyadic formalism is impossible within the framework of non-extensive statistical mechanics.

Highlights

A breakthrough in the field came with the connection of kappa distributions with statistical mechanics, and with the concept of non-extensive statistical mechanics
Showed that the consistent connection of the theory and formalism of kappa distributions with non-extensive statistical mechanics is based on four fundamental physical notions and concepts: (i) q-deformed exponential and logarithm functions [97,98]; (ii) escort probability distribution [99]; (iii) Tsallis entropy [100]; and (iv) physical temperature [101,102,103]
The modern theory of non-extensive statistics is based on the generalization of two elements, fundamental for statistical mechanics, (i) the classical formulation of Boltzmann-Gibbs entropy, and (ii) the notion of canonical distribution via the formalism of ordinary/escort probabilities;

Summary

Introduction

Kappa distributions were employed to describe numerous space plasma populations in: (i) the inner heliosphere, including solar wind [1,2,3,4,5,6,7,8,9,10,11,12,13], solar spectra [14,15], solar corona [16,17,18,19], solar energetic particles [20,21], corotating interaction regions [22], and related solar flares [23,24,25,26]; (ii) the planetary magnetospheres including the magnetosheath [27,28], near the magnetopause [29], magnetotail [30], ring current [31], plasma sheet [32,33,34], ionosphere [35], magnetospheric substorms [36], magnetospheres of giant planets, such as Jovian [37,38], Saturnian [39,40,41], Uranian [42], Neptunian [43], magnetospheres of planetary moons, such as Io [44] and Enceladus [45], or cometary magnetospheres [46,47];(iii) the outer heliosphere and the inner heliosheath [7,8,9,10,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]; (iv) beyond the heliosphere, including HII regions [63], planetary nebula [64,65], and supernova magnetospheres [66]; or other space plasma-related analyses [53,54,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90]. Showed that the consistent connection of the theory and formalism of kappa distributions with non-extensive statistical mechanics is based on four fundamental physical notions and concepts: (i) q-deformed exponential and logarithm functions [97,98]; (ii) escort probability distribution [99]; (iii) Tsallis entropy [100]; and (iv) physical temperature [101,102,103] (that is, the actual temperature of the system). The well-known q-exponential function was deduced as a result of the maximization of Tsallis entropy under the constraints of the canonical ensemble.

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Conclusion