Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas

Abstract

Empirical kappa distributions provide a straightforward replacement of the Maxwell distribution for systems out of thermal equilibrium such as space plasmas. Kappa distributions have become increasingly widespread across space physics with the number of relevant publications following, remarkably, an exponential growth rate. However, a breakthrough in the field came with the connection of kappa distributions with the framework of nonextensive statistical mechanics. This introductory paper clarifies fundamental physical concepts and provides mathematical formulations of the theory of kappa distributions, which are a consequence of the connection of kappa distributions with a solid statistical background. Among others, the paper presents the existence of a consistent definition of temperature in systems out of thermal equilibrium described by kappa distributions, the physical meaning of the kappa index, and the formulation of the kappa distribution of a Hamiltonian. In addition, the paper examines the most frequent values of kappa indices in space plasmas. Statistical analysis reveals trends between the characteristic values of density, temperature, and kappa index of space plasmas. Finally, understanding the kinetic interpretation of the temperature as the mean kinetic energy, and of the kappa index as the correlation of kinetic energies, helps to develop all the possible formulations of isotropic/anisotropic kappa distributions.

First evidence of non-Gaussian solar flare EUV spectral line profiles and accelerated non-thermal ion motion

“…From a real-life observational perspective, the line profiles will be indistinguishable from a Gaussian if κ 20. In the high κ index limit, the characteristic width σ 2 κ has the same 2 We note that σ 2 κ in a kappa distribution of this form is not equal to the actual second moment (variance) of the line, but it is related to it by σ 2 = σ 2 κ /(1−3/2κ) (Livadiotis 2015). Hence, the distribution variance can be easily found from a simple change of variables.…”

“…first and second kinds, and forms where either the "temperature" or "thermal velocity" is kappa dependent e.g. Lazar et al (2016), Livadiotis (2015Livadiotis ( , 2014, Livadiotis & McComas (2009), Hellberg et al (2009), Leubner (2004. The physical interpretation of the value of κ obtained from the line fitting, in terms of the ion velocity distribution, therefore depends on the form of the ion kappa distribution used.…”

Modeling anisotropic Maxwell–Jüttner distributions: derivation and properties

“…In the case of anisotropic temperature, the Maxwell distribution becomes anisotropic on the kinetic degrees of freedom, which contribute differently in the internal energy of the system (e.g., solar wind: Olsen and Leer, 1999;Feldman et al, 1975;Pilipp et al, 1987;Phillips and Gosling, 1990;Kasper et al, 2002;Matteini et al, 2007;Štverák et al, 2008 -magnetosphere: Pilipp andMorfil, 1976;Renuka and Viswanathan, 1978;Tsurutani et al, 1982;Sckopke et al, 1990;Gary, 1992;Bavassano Cattaneo et al, 2006;Nishino et al, 2007;Cai et al, 2008;Winglee and Harnett, 2016 also see the corresponding formulations in Krall and Trivelpiece, 1973;Livadiotis and McComas, 2014a;Livadiotis, 2015).…”

Generalised partition functions: inferences on phase space distributions

“…We noted that this has been done in different ways (Livadiotis andMcComas, 2009, 2013;Yoon et al, 2012;Treumann and Baumjohann, 2014b) yielding r + 1 = 5 2 . Normalisation requires integration over the phase space volume.…”

“…2 For a recent compilation and in-depth discussion of the various aspects and applications of the kappa distribution the reader is referred to the extended presentations contained in Livadiotis andMcComas (2009, 2013) as well as to the almost complete list of papers referenced therein. This list gives a historical record of the work done on and application of the kappa distribution as well as its relation to the celebrated Tsallis nonextensive thermostatistics (Tsallis, 1988;Tsallis et al, 1998;Gell-Mann and Tsallis, 2004). and Baumjohann, 2014a; Treumann and Baumjohann, 2015) of various mathematical functions and functional transforms (see also the reference lists of papers cited in footnote 2).…”