The κ-cookbook: a novel generalizing approach to unify κ-like distributions for plasma particle modelling

Abstract

<jats:title>Abstract</jats:title> <jats:p>In the literature different so-called κ-distribution functions are discussed to fit and model the velocity (or energy) distributions of solar wind species, pickup ions or magnetospheric particles. Here we introduce a generalized (isotropic) κ-distribution as a ”cookbook”, which admits as special cases, or ”recipes”, all the other known versions of κ-models. A detailed analysis of the generalized distribution function is performed, providing general analytical expressions for the velocity moments, Debye length, and entropy, and pointing out a series of general requirements that plasma distribution functions should satisfy. From a contrasting analysis of the recipes found in the literature, we show that all of them lead to almost the same macroscopic parameters with a small standard deviation between them. However, one of these recipes called the regularized κ-distribution provides a functional alternative for macroscopic parameterization without any constraint for the power-law exponent κ.</jats:p>

Olbert’s Kappa Fermi and Bose Distributions

“…(Note that there is no restriction on κ > 0 except that it has to be positive. The analyticity of the distribution, its character of a distribution, and its correct thermodynamic properties are taken care of by s in the exponent, a fixed constant number whose particular nonrelativistic and relativistic values are determined by the thermodynamic constraints [for proof, see, e.g., 23,26,35] and are given below.) It is not a problem to include a high-energy cutoff exp(−ϵ α /ϵ c ) with βϵ c ≫ 1, which truncates the Olbertian at high energies.…”

“…Interpreting κT as physical temperature would be as wrong as it was in the classical case [3]. It has been demonstrated elsewhere [cf., e.g., 23,25,26,31,35] that T remains the real physical temperature in the classical case, a conclusion which is not violated in the quantum domain as its physical meaning is not changed by the transition from classical to quantum physics. For ideal nonrelativistic and relativistic gases, one in addition has s 5/2 and s 4 [12,35], respectively.…”

Olbertian Partition Function in Scalar Field Theory

“…The latter leads directly to the Olbert distribution; the former constructs a similar distribution in suitable approximation. For a collection of other properties of the Olbert distribution, the reader is directed to the extended literature on κ distributions (cf., e.g., [15,16]; and references therein).…”

“…where E α is the energy in the state α, βE c ≫ 1 is some high-energy cutoff [16,30,31], and κ, s are exponents, of which r is free to choose for satisfying thermodynamic needs. This partition function holds for discrete particles occupying the energy states E α .…”

“…They were also derived in plasma wave-wave interaction theory [11][12][13]. Physically, they represent quasi-stationary states far from equilibrium [14][15][16][17][18]. To some degree, they are related to Tsallis' thermostatistics [19].…”