Abstract
In this paper we examine the physical foundations and theoretical development of the kappa distribution, which arises naturally from non-extensive Statistical Mechanics. The kappa distribution provides a straightforward replacement for the Maxwell distribution when dealing with systems in stationary states out of thermal equilibrium, commonly found in space and astrophysical plasmas. Prior studies have used a variety of inconsistent, and sometimes incorrect, formulations, which have led to significant confusion about these distributions. Therefore, in this study, we start from the N-particle phase space distribution and develop seven formulations for kappa distributions that range from the most general to several specialized versions that can be directly used with common types of space data. Collectively, these formulations and their guidelines provide a “toolbox” of useful and statistically well-grounded equations for future space physics analyses that seek to apply kappa distributions in data analysis, simulations, modeling, theory, and other work.
Highlights
Boltzmann-Gibbs (BG) Statistical Mechanics has withstood the test of time for describing classical systems in thermal equilibrium—a state where any flow of heat is in balance
Maxwell distributions are quite rare in space and astrophysical plasmas (e.g. Hammond et al 1996); instead, the vast majority of these plasmas reside in stationary states, that are typically not well described by Maxwell distributions, and not in thermal equilibrium
– In space physics the velocities are typically used to describe the phase space, instead of the standard use of momentums {mun}. – Non-extensive Statistical Mechanics requires that the Hamiltonian function is included such that the internal energy, i.e., the average Hamiltonian H, is normalized to zero in the canonical distribution, i.e., H ({rn}, {wn}) → H ({rn}, {wn}) − H
Summary
Boltzmann-Gibbs (BG) Statistical Mechanics has withstood the test of time for describing classical systems in thermal equilibrium—a state where any flow of heat (e.g. thermal conduction, thermal radiation) is in balance. Space plasmas show strong collective behavior that characterizes the correlated particles within a Debye sphere, without localized phenomena between individual particles due to interactions or collisions. This behavior leads these systems to exotic statistical states that cannot be understood by the classical statistical description of thermal equilibrium. 3 we develop seven kappa distribution formulations, which comprise the basic toolbox for describing the statistics of systems out of thermal equilibrium; these range from the most general to several specialized versions that can be directly used with common types of space data. Appendix B develop and prove the mathematical formulations provided in the toolbox
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