Stochastic dynamics and superstatistics of the many-particle kappa distribution

The diffusion of particles with kappa distributed velocities is strongly influenced by statistical correlations. We argue that the consistent way to deduce the diffusion laws of any one degree of freedom is to analyze the simultaneous diffusion of virtually infinite correlated degrees of freedom. This is done by deriving the diffusion laws (I) by utilizing the superstatistics interpretation of the kappa distribution and averaging the usual Brownian motions correlators over the super-ensemble of fluctuating temperatures, (II) through the one degree of freedom Langevin equation, (III) through the many degrees of freedom Langevin equation, calculating the diffusion of any one degree of freedom. It turns out that only the results (I) and (III) agree. The disagreement between (II) and (III) is a striking outcome of the strong correlations between kappa distributed degrees of freedom. The agreement between (I) and (III) shows that the superstatistics is a fundamental interpretation of the kappa distribution. The discrepancy of (II) shows that focusing on a single degree of freedom or particle is inconsistent with a superstatistics interpretation. Derivation (III) explicitly realizes the recent observation by the authors that the mean energy per degree of freedom is the superstatistical fluctuating temperature in a system with a large number of particles. We conclude that superstatistics is intimately related to a system of correlated degrees of freedom (in our case, kappa distributed); one cannot consistently reason with a single degree of freedom