On the Simplification of Statistical Mechanics for Space Plasmas

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.

Kappa Distributions: Statistical Physics and Thermodynamics of Space and Astrophysical Plasmas

“…where the auxiliary notion of entropy S is connected with the actual entropy S through [30]. However, the temperature is not the only intensive thermodynamic parameter characterizing particle systems.…”

“…In this paper, we show recent developments on the theory of kappa distribution, with emphasis on the differential equations leading to kappa distributions; we also show that the differential equation taken from the Fokker-Planck equation model of Shizgal (2018) [29], and from the earlier first principles analysis of Livadiotis (2018) [30], are equivalent. In Section 2, we briefly present the general theory of kappa distributions, and then show their connection to statistical mechanics; for the first time, this is shown within the framework of continuous description and using the kappa index formalism.…”

“…However, the maximization of entropy can follow an alternative path, in which there is no use for the formalism of escort distributions. In particular, it was shown that this dyadic formalism of distributions can be avoided in order to simplify the theory, but it leads to a dyadic formulation of entropy [30]. According to this, the constraint of internal energy is written as…”

“…We note that we avoid the concept of escort distribution but the kappa index transformation, κ → κ in Equation (22), leads to a duality of entropy, as shown in [30]-that is, S(κ) ↔ S( κ) . It is also important to mention that this alternative method of maximization has the advantage that it can be used for any entropic formulae, e.g., in theoretical analyses that seek to retrieve the entropic form considering other first physical principles (Section 3).…”

Response to Reviewer No_3

“…Empirical kappa distributions were introduced in mid-1960s by Binsack (1966), Olbert (1968), and Vasyliũnas (1968), while their connection with statistical mechanics was shown and studied in detail about half a century later (see Livadiotis and McComas, 2009, and references therein). In particular, the statistical origin of these distributions is now widely accepted to be determined within the framework of nonextensive statistical mechanics (Tsallis, 2009).…”