Kappa and other nonequilibrium distributions from the Fokker-Planck equation and the relationship to Tsallis entropy

Non-Thermal Solar Wind Electron Velocity Distribution Function

The quiet-time solar wind electrons feature non-thermal characteristics when viewed from the perspective of their velocity distribution functions. They typically have an appearance of being composed of a denser thermal "core" population plus a tenuous energetic "halo" population. At first, such a feature was empirically fitted with the kappa velocity space distribution function, but ever since the ground-breaking work by Tsallis, the space physics community has embraced the potential implication of the kappa distribution as reflecting the non-extensive nature of the space plasma. From the viewpoint of microscopic plasma theory, the formation of the non-thermal electron velocity distribution function can be interpreted in terms of the plasma being in a state of turbulent quasi-equilibrium. Such a finding brings forth the possible existence of a profound inter-relationship between the non-extensive statistical state and the turbulent quasi-equilibrium state. The present paper further develops the idea of solar wind electrons being in the turbulent equilibrium, but, unlike the previous model, which involves the electrostatic turbulence near the plasma oscillation frequency (i.e., Langmuir turbulence), the present paper considers the impact of transverse electromagnetic turbulence, particularly, the turbulence in the whistler-mode frequency range. It is found that the coupling of spontaneously emitted thermal fluctuations and the background turbulence leads to the formation of a non-thermal electron velocity distribution function of the type observed in the solar wind during quiet times. This demonstrates that the whistler-range turbulence represents an alternative mechanism for producing the kappa-like non-thermal distribution, especially close to the Sun and in the near-Earth space environment.

Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy

Nonequilibrium systems in chemistry and physics are generally modeled with the Boltzmann, Fokker-Planck, and Master equations. There has been a considerable interest in the nonequilibrium distributions of electrons and ions in space physics in different environments as well as in other systems. An often-used empirical model to characterize these distributions, especially in space physics, is the Kappa distribution. There have been numerous efforts to provide a theoretical basis for the Kappa distribution that include the Fokker-Planck equation with specific drift and diffusion coefficients. Alternatively, the maximization of the Tsallis nonextensive entropy provides the desired Kappa distribution. This paper examines three families of Fokker-Planck equations that provide a steady-state Kappa distribution as well as a myriad of other nonequilibrium distributions. The relationship of these works with analogous studies of distributions with asymptotic high-energy tails is also considered. It is clear that the many different nonequilibrium distribution functions that can occur cannot all be rationalized with Gibbs-Boltzmann statistical mechanics, which uniquely gives equilibrium distributions, or with the Tsallis nonextensive entropy, which gives uniquely the Kappa distribution. The current research is directed towards an improved understanding of the origin of nonequilibrium distributions in several specific systems.

Thermodynamic Definitions of Temperature and Kappa and Introduction of the Entropy Defect

This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the "entropy defect" in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm.

Fokker-Planck equation for Coulomb relaxation and wave-particle diffusion: Spectral solution and the stability of the Kappa distribution to Coulomb collisions

The present paper considers the time evolution of a charged test particle of mass m in a constant temperature heat bath of a second charged particle of mass M. The time dependence of the distribution function of the test particles is given by a Fokker-Planck equation with a diffusion coefficient for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. For the mass ratio m/M→0, the steady distribution is a Kappa distribution which has been employed in space physics to fit observed particle energy spectra. The time dependence of the distribution functions with some initial value is expressed in terms of the eigenvalues and eigenfunctions of the linear Fokker-Planck operator and also interpreted with the transformation to a Schrödinger equation. We also consider the explicit time dependence of the distribution function with a discretization of the Fokker-Planck equation. We study the stability of the Kappa distribution to Coulomb collisions.

Thermodynamic, Non-Extensive, or Turbulent Quasi-Equilibrium for the Space Plasma Environment

The Boltzmann–Gibbs (BG) entropy has been used in a wide variety of problems for more than a century. It is well known that BG entropy is additive and extensive, but for certain systems such as those dictated by long-range interactions, it is speculated that the entropy must be non-additive and non-extensive. Tsallis entropy possesses these characteristics, and is parameterized by a variable q (q=1 being the classic BG limit), but unless q is determined from microscopic dynamics, the model remains a phenomenological tool. To this day, very few examples have emerged in which q can be computed from first principles. This paper shows that the space plasma environment, which is governed by long-range collective electromagnetic interaction, represents a perfect example for which the q parameter can be computed from microphysics. By taking the electron velocity distribution function measured in the heliospheric environment into account, and considering them to be in a quasi-equilibrium state with electrostatic turbulence known as quasi-thermal noise, it is shown that the value corresponding to q=9/13=0.6923, or alternatively q=5/9=0.5556, may be deduced. This prediction is verified against observations made by spacecraft, and it is shown to be in excellent agreement. This paper constitutes an overview of recent developments regarding the non-equilibrium statistical mechanical approach to understanding the non-extensive nature of space plasma, although some recent new developments are also discussed.

Electron Energy Partition across Interplanetary Shocks. I. Methodology and Data Product

Analyses of 15,314 electron velocity distribution functions (VDFs) within ±2 hr of 52 interplanetary (IP) shocks observed by the Wind spacecraft near 1 au are introduced. The electron VDFs are fit to the sum of three model functions for the cold dense core, hot tenuous halo, and field-aligned beam/strahl component. The best results were found by modeling the core as either a bi-kappa or a symmetric (or asymmetric) bi-self-similar VDF, while both the halo and beam/strahl components were best fit to bi-kappa VDF. This is the first statistical study to show that the core electron distribution is better fit to a self-similar VDF than a bi-Maxwellian under all conditions. The self-similar distribution deviation from a Maxwellian is a measure of inelasticity in particle scattering from waves and/or turbulence. The ranges of values defined by the lower and upper quartiles for the kappa exponents are κ ec ~ 5.40-10.2 for the core, κ eh ~ 3.58-5.34 for the halo, and κ eb ~ 3.40-5.16 for the beam/strahl. The lower-to-upper quartile range of symmetric bi-self-similar core exponents is s ec ~ 2.00-2.04, and those of asymmetric bi-self-similar core exponents are p ec ~ 2.20-4.00 for the parallel exponent and q ec ~ 2.00-2.46 for the perpendicular exponent. The nuanced details of the fit procedure and description of resulting data product are also presented. The statistics and detailed analysis of the results are presented in Paper II and Paper III of this three-part study.

Kappa Distributions: Statistical Physics and Thermodynamics of Space and Astrophysical Plasmas

Kappa distributions received impetus as they provide efficient modelling of the observed particle distributions in space and astrophysical plasmas throughout the heliosphere. This paper presents (i) the connection of kappa distributions with statistical mechanics, by maximizing the associated q-entropy under the constraints of the canonical ensemble within the framework of continuous description; (ii) the derivation of q-entropy from first principles that characterize space plasmas, the additivity of energy, and entropy; and (iii) the derivation of the characteristic first order differential equation, whose solution is the kappa distribution function.