The plasma particle velocity distributions observed in the solar wind generally show enhanced (non-Maxwellian) suprathermal tails, decreasing as a power law of the velocity and well described by the family of Kappa distribution functions. The presence of non-thermal populations at different altitudes in space plasmas suggests a universal mechanism for their creation and important consequences concerning plasma fluctuations, the resonant and nonresonant wave - particle acceleration and plasma heating. These effects are well described by the kinetic approaches where no closure requires the distributions to be nearly Maxwellian. This paper summarizes and analyzes the various theories proposed for the Kappa distributions and their valuable applications in coronal and space plasmas.

Kappa Distributions: Theory and Applications in Space Plasmas

[1] Empirically derived kappa distributions are becoming increasingly widespread in space physics as the power law nature of various suprathermal tails is melded with more classical quasi-Maxwellian cores. Two different mathematical definitions of kappa distributions are commonly used and various authors characterize the power law nature of suprathermal tails in different ways. In this study we examine how kappa distributions arise naturally from Tsallis statistical mechanics, which provides a solid theoretical basis for describing and analyzing complex systems out of equilibrium. This analysis exposes the possible values of kappa, which are strictly limited to certain ranges. We also develop the concept of temperature out of equilibrium, which differs significantly from the classical equilibrium temperature. This analysis clarifies which of the kappa distributions has primacy and, using this distribution, the kinetic and physical temperatures become one, both in and out of equilibrium. Finally, we extract the general relation between both types of kappa distributions and the spectral indices commonly used to parameterize space plasmas. With this relation, it is straightforward to compare both spectral indices from various space physics observations, models, and theoretical studies that use kappa distributions on a consistent footing that minimizes the chances for misinterpretation and error. Now that the connection is complete between empirically derived kappa distributions and Tsallis statistical mechanics, the full strength and capability of Tsallis statistical tools are available to the space physics community for analyzing and understanding the kappa-like properties of the various particle and energy distributions observed in space.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2009JA014352

Beyond kappa distributions: Exploiting Tsallis statistical mechanics in space plasmas

In this paper we examine the physical foundations and theoretical development of the kappa distribution, which arises naturally from non-extensive Statistical Mechanics. The kappa distribution provides a straightforward replacement for the Maxwell distribution when dealing with systems in stationary states out of thermal equilibrium, commonly found in space and astrophysical plasmas. Prior studies have used a variety of inconsistent, and sometimes incorrect, formulations, which have led to significant confusion about these distributions. Therefore, in this study, we start from the N-particle phase space distribution and develop seven formulations for kappa distributions that range from the most general to several specialized versions that can be directly used with common types of space data. Collectively, these formulations and their guidelines provide a “toolbox” of useful and statistically well-grounded equations for future space physics analyses that seek to apply kappa distributions in data analysis, simulations, modeling, theory, and other work.

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Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics

Recent advances in Space Physics theory have shown the connection between non-extensive Statistical Mechanics and space plasmas by providing a theoretical basis for the empirically derived kappa distributions commonly used to describe the phase-space distribution functions of these systems. The non-equilibrium temperature and the kappa index that govern these distributions are the two independent controlling parameters of non-equilibrium systems. The significance of the kappa index is primarily given by its role in identifying the non-equilibrium stationary states and measuring their "thermodynamic distance" from thermal equilibrium, while its physical meaning is connected to the correlation between the system's particles. The classical, single stationary state at equilibrium is generalized into a whole set of different non-equilibrium stationary states labeled by the kappa index. This paper addresses certain crucial issues about the physical meaning and role of the kappa index in identifying stationary states. The origin of the emerged inconsistencies is that the kappa index is not an invariant physical quantity, but instead depends on the degrees of freedom of the system's particles. This leads in several misleading conclusions, such as (1) only large kappa index, practically infinite, can characterize the many-particle kappa distribution, and (2) the correlation between particles depends on the total number of the system's particles. Here we show that a modified kappa index, invariant for any number of degrees of freedom, can be naturally defined. Then, we develop and examine the relevant corrected formulation of many-particle multidimensional kappa distribution, and discuss the physical meaning of the invariant kappa index.

https://iopscience.iop.org/article/10.1088/0004-637X/741/2/88/pdf

Invariant kappa distribution in space plasmas out of equilibrium

Space plasmas from the solar wind to planetary magnetospheres and the outer heliosphere are systems in stationary states out of equilibrium. Empirical kappa distributions, which naturally emerge from Tsallis Statistics, successfully describe these space plasmas. The Tsallis formalism offers a solid statistical foundation and provides a set of proven tools for understanding these distributions, including a consistent definition of temperature-the physical temperature, which characterizes the non-equilibrium stationary states. Here, we develop a measure of the 'thermodynamic distance' of stationary states away from equilibrium. The stationary states are labeled by the value of the entropic q-index, lying in a spectrum from q = 1 (equilibrium) to the maximum value of q, which specifies the furthest possible stationary state from equilibrium. We call this the 'q-frozen state', because as a system approaches this state, it behaves analogously to when its temperature approaches absolute zero. We also introduce a novel isothermal procedure that describes a system's transition into different stationary states by varying the q-index, and show how the variation of temperature can be realized using an 'iso-metastability' procedure, in which the system remains in a fixed stationary state. These innovations allow a generalization of the zeroth law of thermodynamics to cover stationarymore » states out of equilibrium. By expressing the entropy in terms of the q-index, we show the detailed paths by which the transition of stationary states evolves toward equilibrium following the dynamics of a characteristic difference equation along the q-indices. This naturally exhibits certain stationary states out of equilibrium that are frequently observed in space plasmas.« less

https://iopscience.iop.org/article/10.1088/0004-637X/714/1/971/pdf

Exploring Transitions of Space Plasmas Out of Equilibrium

. In usual incoherent scatter data analysis, the plasma distribution function is assumed to be Maxwellian. In space plasmas, however, distribution functions with a high energy tail which can be well modeled by a generalized Lorentzian distribution function with spectral index kappa (kappa distribution) have been observed. We have theoretically calculated incoherent scatter spectra for a plasma that consists of electrons with kappa distribution function and ions with Maxwellian neglecting the effects of the magnetic field and collisions. The ion line spectra have a double-humped shape similar to those from a Maxwellian plasma. The electron temperatures are underestimated, however, by up to 40% when interpreted assuming Maxwellian distribution. Ion temperatures and electron densities are affected little. Accordingly, actual electron temperatures might be underestimated when an energy input maintaining a high energy tail exists. We have also calculated plasma lines with the kappa distribution function. They are enhanced in total strength, and the peak frequencies appear to be slightly shifted to the transmitter frequency compared to the peak frequencies for a Maxwellian distribution. The damping rate depends on the electron temperature. For lower electron temperatures, plasma lines for electrons with a κ distribution function are more strongly damped than for a Maxwellian distribution. For higher electron temperatures, however, they have a relatively sharp peak. Key words: Ionosphere (auroral ionosphere; plasma waves and instabilities) – Space plasma physics (kinetic and MHD theory)

/pdf/effects-of-a-kappa-distribution-function-of-electrons-on-3b4ulbbf0f.pdf

Effects of a kappa distribution function of electrons on incoherent scatter spectra

Using EISCAT radar data, we find that electrons are strongly heated in the magnetic field-line direction during high electric field events. The remote site data show that the electron temperature increases in almost the same way in the field-perpendicular direction; electron heating by E region plasma turbulence is isotropic. We discuss the implications of our observation for the "plasmon"-electron as well as the wave Joule heating models of the anomalous electron heating in the E region.

/pdf/observation-of-isotropic-electron-temperature-in-the-26s8tchhxv.pdf

S. Saito

Papers

Effects of a kappa distribution function of electrons on incoherent scatter spectra

Observation of isotropic electron temperature in the turbulent E region