Abstract

• Many space and laboratory plasmas are found to possess non-Maxwellian distribution functions. An empirical function promoted by Stan Olbert, which superposes a Maxwellian core with a power-law tail, has been found to emulate many of the plasma distributions discovered in space. These $\kappa$-functions, with their associatedpower-law tail induced anomalous heat flux, have been used by theorists$^1$\ as the origin of solar coronal heating of solar wind. However, the principle and prerequisite for the robust production of such a non-equilibrium distribution has rarely been explained. We report on recent statistical work$^2$, which shows that the $\kappa$-function is one of a general class of solutions to a time-fractional diffusion equation, known as a L\'evy stable probability distribution. These solutions arise from time-variable probability distribution (or equivalently, a spatially variable probability in a flowing medium), which demonstrate that anomalously high flux, or equivalently, non-equilibrium thermodynamics govern the outflowing solar wind plasma. We will characterize the parameters that control the degree of deviation from a Maxwellian and attempt to draw physical meaning from the mathematical formalism. $^1$Scudder, J. {\it Astrophys. J.}, 1992.\$^2$Mainardi, F. and R. Gorenflo, {\it J. Computational and Appl. Mathematics, Vol. 118}, No 1-2, 283-299 (2000).