Turbulent Equilibrium and Nonextensive Entropy

Abstract

The Boltzmann-Gibbs (BG) entropy has been used in a wide variety of problems for almost 130 years. It is well known that BG entropy is extensive. However, for systems dictated by long-range forces, such as plasmas, the entropy must be non-extensive. Over the years, attempts were made to generalize BG entropy to non-extensive systems. Of these, Tsallis entropy characterized by the q parameter (q = 1 being the BG limit) has gained popularity. However, unless q is determined from microscopic dynamics, the model remains a phenomenological tool. To this date very few examples have emerged in which q can be computed from first principles. The present paper demonstrates that for electrons in dynamic equilibrium with steady-state Langmuir turbulence, the q parameter may be computed on the basis of plasma equations. It will also be shown that the steady-state electrons are characterized by a kappa-like velocity distribution, which resembles the most probable state in the Tsallis thermostatics. This strongly suggests that the quasi-equilibrium between Langmuir turbulence and electrons may be characterized by non-extensive entropy concept.