Kappa distributions in the framework of superstatistics

TL;DR

Problem: explain the origin of kappa velocity distributions in collisionless plasmas within a Boltzmann-Gibbs framework. Approach: derive both multi-particle and single-particle kappa distributions from a scale-invariant gamma model for inverse temperature within the superstatistical formalism, and compute energy moments, kinetic-energy correlations, observable temperature, and entropy. Key findings: the N-particle distribution is $P(\mathbf{V}|u,\beta_S) \propto [1 + u\beta_S K(\mathbf{V})]^{-(1/u+3N/2)}$ with $\kappa_N = 1/u + 3N/2 - 1$; moments $\langle K^n \rangle_{u,\beta_S} = (u\beta_S)^{-n} \frac{\Gamma(3N/2+n)}{\Gamma(3N/2)} \frac{\Gamma(1/u - n)}{\Gamma(1/u)}$; inter-particle correlations yield $\rho_{k_1,k_2}(u) = 3u/(u+2)$ and mutual information $I_{k_1,k_2}(u)$ increasing with $u$; the conditional inverse-temperature distribution remains gamma with mean and vanishing relative variance in the thermodynamic limit; entropy decomposes into beta-uncertainty, velocity, and correlation components and becomes extensive per particle as $N\to\infty$. Significance: provides a rigorous, Boltzmann-Gibbs–consistent route to kappa plasmas, clarifies the role of temperature fluctuations on observables, and offers generalizable tools for other superstatistics families.

Abstract

The kappa distribution of velocities appears routinely in the study of collisionless plasmas present in Earth's magnetosphere, the solar wind among other contexts where particles are unable to reach thermal equilibrium. Originally justified through the use of Tsallis' non-extensive statistics, nowadays there are alternative frameworks that provide insight into these distributions, such as superstatistics. In this work we review the derivation of the multi-particle and single-particle kappa distributions for collisionless plasmas within the framework of superstatistics, as an alternative to the use of non-extensive statistics. We also show the utility of the superstatistical framework in the computation of expectation values under kappa distributions. Some consequences of the superstatistical formalism regarding correlations, temperature and entropy of kappa-distributed plasmas are also discussed.

Figures (4)

• Figure 1: Left, skewness of the kinetic energy distribution as a function of $u$. Right, kurtosis of the same distribution of kinetic energies.
• Figure 2: Kinetic energy distribution for $\beta_S = 1$ and different values of $u$ between 0 and 1/2, according to (\ref{['eq:marg_k1']}).
• Figure 3: Left, mutual information $I_{k_1, k_2}(u)$ in (\ref{['eq:mutual']}) as a function of $u$. Right, Pearson correlation $\rho_{k_1, k_2}(u)$ in (\ref{['eq:pearson']}) as a function of $u$.
• Figure 4: Left, joint entropy $\mathcal{S}_{\beta, \bm V}$ and superstatistical entropy $\mathcal{S}_{\beta}$ as a function of $u$ for several values of $\beta_S$. Right, entropy difference $\mathcal{S}_{\text{corr}}$ associated to the correlation between $\beta$ and $\bm V$, as a function of $u$.