Chapter 6 - Superstatistics: Superposition of Maxwell–Boltzmann Distributions

Abstract

In this chapter we think about dynamical reasons why kappa distribution occur so frequently in plasmas and other systems. In fact many complex driven nonequilibrium systems in statistical physics and in other areas of science (including plasma physics) are effectively described by a superposition of several statistics on different timescales, in short “superstatistics.” A simple example is a tracer particle moving in a spatially inhomogeneous medium with temperature fluctuations on a large scale or some other parameter fluctuations on a large scale, but the concept is much more general. Superstatistical systems typically have marginal distributions that exhibit fat tails, for example, power law tails or stretched exponentials. Kappa distributions in space plasma physics are a particular example. In most cases observed for a variety of applications, one finds that there are three relevant universality classes: log-normal superstatistics, chi-square superstatistics, and inverse chi-square superstatistics. These can be effectively described by methods of nonextensive statistical mechanics. We outline the underlying theory; comment on the timescale separation assumption, on universality aspects, and the behavior of the tails of the distributions; and we briefly describe how to extract superstatistical parameters from a measured experimental time series. We finally describe some examples where kappa distributions (and their superstatistical descriptions) play an important role, namely, classical and quantum turbulence as well as scattering processes in high-energy physics.