Kappa distributions in the framework of superstatistics

A Kappa distribution function applicable to systems comprising mixed fermions and bosons has been developed through the thermodynamic Gibbs potential, utilizing the quantum versions of Olbert’s Kappa distributions. The generalized expressions of the partition function and the entropy have been evaluated for such mixed quantum systems. The analysis shows that boson-rich systems consistently exhibit higher entropy than fermion-rich systems. The distribution functions show heavy-tailed characteristics at low Kappa index values, indicating the presence of superthermal particles. It is observed that relativistic effects lead to a significant increase in entropy.

Abstract Solar flares effectively accelerate particles to nonthermal energies. These accelerated electrons are responsible for energy transport and subsequent emissions in hard X-ray (HXR), radio, and UV/EUV radiation. Due to the steeply decreasing electron spectrum, the electron population and consequently the overall flare energetics are predominantly influenced by low-energy nonthermal electrons. However, deducing the electron distribution in this energy-containing range remains a significant challenge. In this study, we apply the warm-target HXR emission model with kappa-form injected electrons to two well-observed GOES M-class flares. Moreover, we utilize EUV observations to constrain the flaring plasma properties, which enables us to determine the characteristics of accelerated electrons across a range from a few keV to tens of keV. We demonstrate that the warm-target model (WTM) reliably constrains the properties of flare-associated electrons, even accounting for the uncertainties that had previously been unaddressed. The application of a kappa distribution for the accelerated electrons allows for meaningful comparisons with electron distributions inferred from EUV observations, specifically for energy ranges below the detection threshold of RHESSI. Our results indicate that the accelerated electrons constitute only a small fraction of the total electron population within the flaring region. Moreover, the physical parameters, such as electron escape time and acceleration timescale, inferred from both the WTM and the EUV observations further support the scenario in which electrons undergo thermalization within the corona. This study highlights the effectiveness of integrating the WTM with EUV observations to accurately characterize energy-containing electrons and their associated acceleration and transport processes.

Abstract. This work derives transport coefficients, i.e., electrical conductivity, thermoelectric, diffusion, and mobility coefficients, for a Lorentz plasma with a modified Kappa distribution. The derivation begins by formulating transport equations (continuity, momentum, and energy) within the five-moment approximation, using the modified Kappa distribution as the zeroth-order function. Subsequently, the corresponding momentum and energy collision terms are evaluated via the Boltzmann collision integral for different types of collisions, including Coulomb collisions, hard-sphere interactions, and Maxwell molecule collisions. Next, we use the momentum equation from the five-moment approximation to obtain the generalized Ohm’s law and extended Fick's law, leading to the transport coefficients. Furthermore, the influence of the kappa parameter on the collision terms and transport coefficients is analyzed. The traditional results based on the Maxwellian distribution are recovered in the limit as kappa parameter approaches infinity.

This paper presents the first Vlasov simulations of whistler mode waves involving the subtracted kappa distribution. This type of distribution is a generalization of the subtracted Maxwellian involving a loss cone as well as a nonthermal energetic tail controlled by the index called κ. The large index κ transforms the subtracted kappa distribution to a subtracted Maxwellian distribution. The simulation shows that the nonthermal features of the subtracted kappa distribution excite whistler wave to a higher nonlinear state compared to the subtracted Maxwellian distribution. The variation of the saturated state is examined with the loss cone parameters for different values of spectral index κ. The study demonstrates that the growth of whistler instability diminishes as the loss cone becomes populated with additional particles. In contrast, a notable increase in instability growth is observed when the slope of the distribution function steepens. These results remain consistent for both the subtracted kappa and subtracted Maxwellian distributions within the framework of Vlasov theory and simulations, highlighting the critical factors that influence whistler instability in nonthermal plasmas.

Abstract For kinetic modeling of plasma processes in space, a rejection-sampling procedure for generating a Kappa distribution in particle-in-cell simulation is proposed. A Pareto distribution is employed as an envelope distribution. The procedure only requires uniform variates, and its acceptance efficiency is ≈0.73–0.8.

The generalized fluctuation–dissipation relations that produce the regularized Kappa distributions are studied. The two-variable Fokker–Planck equation, as well as its reductions in the absence of potential and in the overdamped limit, are considered. All these Fokker–Planck equations have the regularized Kappa distributions as the stationary solutions if the friction and diffusion coefficients satisfy the generalized fluctuation–dissipation relations. In addition, we prove that the principle of detailed balance holds for all the stationary solutions derived in this work.

Abstract Space plasmas have been widely observed to have a non-Maxwellian distribution function with high-energy tails. One of the most used ways to describe this type of plasma is through Kappa distributions. These distributions have been found to introduce several non-trivial changes to plasma dynamics, particularly to the properties and evolution of plasma waves. In this article, using the kinetic theory of plasma waves, we characterize the critical angle of total internal reflection in a non-magnetized Kappa-distributed plasma. Considering Kappa distributions, we have obtained expressions for the permittivity of the medium, the refractive index, and the total internal reflection critical angle as functions of the wave number, the parameter κ , temperature, and density. Our results show that the relative kappa value and the temperature of the media play an important role in determining the critical angle. Moreover, we have found that the critical angle as a function of wave number exhibits a minimum value when the wavelength is equal to the inertial length. These results demonstrate that kinetic effects are relevant for understanding total internal reflection, particularly in elucidating the transformations that electromagnetic waves undergo when transitioning from one region to another in space environments.

Abstract We present a detailed linear analysis of right-handed (RH) proton instabilities in beaming bi-Maxwellian/bi-Kappa plasma representative of young solar wind conditions currently explored by the Parker Solar Probe. At lower heliocentric distances, ion (proton) beams are more dense, and the associated wave fluctuations are predominantly RH polarized, strongly suggesting the presence of proton-beam plasma instabilities to explain not only the enhanced fluctuations but also the subsequent particle diffusion in energy and pitch angle. The study systematically investigates the cumulative effects of beam drift velocity and temperature anisotropies on the ion–ion resonant and proton firehose instabilities. By examining the variations in maximum growth rates, resonant factors, and instability thresholds across a broad range of plasma parameters, including beam and core anisotropies, density ratios, plasma beta, and suprathermal populations, this work identifies the key regimes where instabilities are enhanced or suppressed. The influence of suprathermal ions and electrons, modeled using Kappa distributions, is shown to significantly extend the unstable parameter space. Electron temperature anisotropy is also found to strongly modify the growth rates of these instabilities by changing their resonance conditions with beaming particles. This parametric study provides comprehensive guidance for future quasilinear or simulation studies by identifying the most physically relevant and numerically effective regimes for these instabilities in the solar wind and space plasmas.

Ganymede’s UV aurorae, observed by HST and Juno/UVS, trace interactions between its atmosphere and Jupiter’s magnetosphere. These emissions, dominated by O I lines at 130.4 and 135.6 nm, are driven by electron impact on species such as H O, O, and O and yet the properties of the precipitating electrons remain poorly constrained. Our aim was to retrieve the energy and flux of precipitating electrons using UV observations from Juno/UVS during PJ34 and to assess the dominant atmospheric species producing the observed emissions. Using the TransPlanet electron transport model and a non-local thermodynamic equilibrium (non-LTE) radiative transfer module, we simulated O I emissions for 17 auroral subregions, testing both monoenergetic and kappa-type electron distributions. The I(135.6 nm )/I(130.4 nm ) line ratio was used as a diagnostic, with values varying by target species. Monoenergetic distributions fit most regions better, with mean energies of 17–300 eV and fluxes up to 2 mW,m . Kappa and Maxwellian distributions yielded higher fluxes, but poorer spectral fits. Poor fits in some regions reflect low S/N or non-ideal electron populations. Our results suggest that Ganymede’s UV aurorae are mainly driven by low- to intermediate-energy electrons. Upcoming high-resolution observations and in situ data from Juice and Europa Clipper will be key to refining these diagnostics.

The magnetosheath of Earth plays a crucial role in shaping the plasma composition of Earth’s magnetosphere. It is instrumental in controlling the thermal properties of magnetospheric plasma and is fundamental to the dynamic processes that govern the evolution of the magnetosphere. Nonlinear solitary waves are frequently observed across diverse regions of Earth’s magnetosphere, utilizing Wideband Data (WBD) plasma wave receivers aboard Cluster Wave Experiment Consortium (WEC). Previous and current space missions have enabled the meticulous observation and analysis of the characteristics of solitary waves, including their amplitude, density, temperature, and temporal durations of waves. In this study, we explore a magnetized plasma system encompassing an inertial ion fluid alongside electron populations characterized by regularized kappa distribution, exhibiting two distinct temperature profiles (cold & hot). Such two distinct electron profiles, alongside data derived from a multitude of observations and laboratory experiments, have been meticulously integrated into the theoretical model presented in this article, aiming to enhance our comprehension of the Earth’s magnetosheath region. A nonlinear equation named Laedke–Spatschek equation is derived using perturbation framework for the plasma system. The examination focuses on propagating plane waves to explore phase-space dynamics. A magnetic field adds complexity to nonlinear wave dynamics, leading to intricate phase-space behaviour. These nonlinear waves exhibit various structures, including chaotic, quasi-periodic (irrational), multi-periodic (rational), and periodic (rational) oscillations, depending on supra-thermal indices and magnetic field strength. Additionally, our research delves into the synergistic effects that arise from the interplay between magnetic field strength, ion concentration level, and the dynamics of high-energy electron populations, all of which contribute to the overall wave dynamics. The findings from our model are anticipated to provide significant insights into not only the Earth’s magnetosheath but also broader astrophysical environments where similar conditions may occur.

The ionospheric Pedersen and Hall conductances play an important role in understanding the coupling by which angular momentum, energy, and matter are exchanged between the magnetosphere, ionosphere and thermosphere at Jupiter, modifying the composition and temperature of the planet. In the high-latitude regions, the Pedersen and Hall conductances are enhanced by the auroral electron precipitation. We investigated the effect of a broadband-precipitating electron energy distribution, similar to the observed electron distributions through particle measurements, on the Pedersen and Hall conductance values. The new conductance values were compared to those obtained from previous studies, notably for a mono-energetic distribution. The broadband-precipitating electron energy distribution was modeled by a kappa distribution, which is used as an input in an electron transport model that computes the vertical density profiles of ionospheric ions. Assuming that the conductivity is mostly governed by the density of and we then evaluated the vertical profiles of the Pedersen and Hall conductivities from the vertical profiles of the ion density. Finally, the Pedersen and Hall conductances were computed by integrating the corresponding conductivities over altitude. The Pedersen and Hall conductance values are globally higher for a broadband electron energy distribution instead of a mono-energetic distribution. In addition, the considered electron collision cross sections and the chosen method for computing the ion production rates can also have significant impacts on the conductance values. The comparison between our results and those deduced from corotation enforcement theory suggests that either a physical mechanism limits the field-aligned currents or the auroral electrons precipitating in the atmosphere are accelerated by processes that are not associated with the field-aligned currents.

We previously investigated the stability threshold of the ion-ion acoustic instability (IIAI) in parameter regimes compatible with recent Parker Solar Probe (PSP) observations, in the presence of a Maxwellian electron distribution. We find that the observed parameters are close to the instability threshold, but IIAI requires a higher electron temperature than what is observed. As electron distributions in the solar wind present clear non-Maxwellian features, we investigated if deviations from the Maxwellian distribution could explain the observed IIAI. We address specifically the kappa (ąppa) and core-strahl distributions for the electrons. We performed analytical studies and kinetic simulations using a Vlasov-Poisson code in a parameter regime relevant to PSP observations. The simulated growth rates were validated against kinetic theory. We show that the IIAI threshold changes in the presence of ąppa or core-strahl electron distributions, but not significantly. In the latter case, simulations confirm the expression of an effective temperature for an equivalent Maxwellian electron distribution. Such an effective temperature could simplify stability assessments of future observations.

Noninvasive direct measurements of higher-order moments of the electron velocity distribution function (EVDF) are needed to improve the understanding of non-Maxwellian electron behavior in various plasmas. This work presents a Bayesian inference method with Monte Carlo sampling to infer the electron heat flux and excess kurtosis from ILTS spectra, which also improves the inferences of the lower-order moments. The method assumes that the EVDFs are described by the sum of at most four super-Gaussians and is tested against synthetic spectra that are representative of ITLS measurements in low-temperature plasmas. Fifteen synthetic spectra are considered that include Maxwellian, Druyvesteyn, and Kappa distributions and their skewed counterparts. For all synthetic spectra considered, the true value of the heat flux is within the uncertainty bounds of the inference. Regarding the excess kurtosis, the true value of the excess kurtosis is within the uncertainty bounds of the inference for all cases except for the Kappa distributions with no or low skewness. At the signal-to-noise ratio of the synthetic spectra, the minimum detectable skewness and excess kurtosis are around ±0.006 and ±0.07, respectively. When the heat flux and excess kurtosis are significantly above their minimum detectable values, relative uncertainties range between 40% and 5%. Finally, in terms of symmetric or low-skewness EVDFs, we find that ILTS is best suited for EVDFs with negative excess kurtosis, suggesting that ILTS can accurately and precisely measure nonequilibrium electron properties in many low-temperature plasmas.

This study investigates the generation and evolution of electromagnetic ion cyclotron (EMIC) waves in the inner magnetosphere by integrating in situ observations with a kinetic theoretical framework based on a kappa-distributed plasma model. Using Cluster spacecraft data from July 18, 2005, the occurrence of EMIC waves is correlated with enhanced cold plasma density and localized magnetic field variations. Ion velocity distribution functions (VDFs) observed during EMIC activity exhibit significant suprathermal tails, justifying the application of kappa distribution fits to model nonthermal proton populations. A dispersion relation for obliquely propagating EMIC waves is derived for such non-Maxwellian conditions, incorporating observed plasma parameters. Numerical solutions of the dispersion relation reveal temporal variations in wave growth rates, with initial intervals showing rapid instability, indicative of local wave generation, and later intervals exhibiting diminished growth, suggestive of wave propagation from remote sources. These results highlight the sensitivity of EMIC wave dynamics to variations in ion anisotropy and suprathermal features of the observed VDFs, thereby contributing to a more comprehensive understanding of wave–particle interactions and energy redistribution processes in Earth's magnetospheric plasma.

Context. In this study, we apply a novel heuristic core-strahlo (CS) model to analyze solar wind electrons. This model reproduces the behavior of a core-halo-strahl representation by employing solely two subpopulations: a bi-Maxwellian core and a modified Kappa distribution that introduces skewness. This modification effectively represents halo and strahl electrons within a single skew distribution. Aims. This work aims to demonstrate that the CS model can be utilized to model observations beyond theoretical contexts. The CS model can reproduce the main features of electron velocity distribution functions (eVDFs) in the solar wind–thermal core, enhanced tails, and skewness–with the advantage that a single parameter controls the asymmetry. Methods. We implemented a comprehensive statistical analysis of solar wind electrons at 1 AU using the electron and solar wind plasma moments on board the NASA Wind SWE/VEIS instrument. This work uses a sophisticated algorithm developed to analyze and characterize separately the core and strahlo populations. We limited our effective energy from 10 eV to 3 keV and fit the eVDFs measurements observed by the WIND satellite to the CS model. Results. Our experimental analysis show good agreement with existing models of solar wind electrons, including those that account for core, halo, and strahl components, as the resulting values fall within the expected order of magnitude. The CS model not only achieves results comparable to previous studies, but also offers the added capability of accounting for heat flux and the asymmetry of the electron velocity distribution through the δ parameter, which enhances our understanding of solar wind electron dynamics. Further, we confirm that the kappa parameter (κ) is independent of the skewness parameter (δ), consistent with previous theoretical studies’ findings. Conclusions. This work serves as an initial practical application of the CS model. We extend its relevance beyond theoretical contexts to the study of observational data. This novel approach not only highlights the specific dynamics of solar wind electrons but also provides insights into their behavior. Specifically, as the strahl relaxes, the halo becomes more enhanced.

Abstract Turbulence in space plasmas remains a fundamental challenge, and Earth's magnetosphere (MSP) offers a natural laboratory for its study. Using high‐resolution magnetic field data from the Magnetospheric Multiscale ( MMS ) mission, we extend a stochastic Markovian framework to analyze turbulence across 10 diverse magnetospheric regions, including key reconnection sites. We show that magnetic field fluctuations are well described as Markov processes in scale across the kinetic domain. Multi‐scale conditional Probability Density Functions (PDFs) reveal Markovian properties beyond the Einstein‐Markov scale. The Kramers‐Moyal coefficients display linear and quadratic dependencies, and the Fokker‐Planck equation accurately reproduces empirical conditional PDFs, with stationary solutions matching observed Kappa distributions. Scale invariance, characterized by power‐law behavior and self‐similar PDFs, holds up to s in most regions but breaks in day‐side reconnection jets. These findings support the universality of the Markovian cascade description across the magnetosphere, while identifying region‐specific deviations that reflect differing energy dissipation mechanisms.

ABSTRACT The study of shock waves in complex plasma systems is crucial for understanding various phenomena, such as black holes, various hurricanes, and industrial applications. The study of the propagation and properties of cylindrical dust acoustic shock waves (CDAShWs) is investigated in a self‐gravitational magnetized dusty plasma system, consisting of warm viscous dust fluid and superthermal electrons and ions represented by a kappa distribution. To study the nonlinear dynamics and collisions of dust acoustic shock waves (DAShWs) in the specified plasma environment, we used theoretical and analytical methods. The magnetohydrodynamic (MHD) plasma model is described by equations of motion. Then, we used the reductive perturbation method to derive the nonlinear Korteweg–de Vries–Burgers (KdV‐B) equation. Our results show that increasing the superthermality index ( κ ) significantly increases the CDAShW amplitude. The properties of CDAShWs are affected by nonplanar cylindrical geometries and various plasma parameters. The structure of CDAShWs and their electric field (EF) is similar to the outer structures observed in black‐hole plasma (BHP), especially in accretion disk plasma and plasma jets. This research enables us to understand wave propagation in complex plasmas by bridging theoretical insights with practical implications. It contributes to a deeper comprehension of plasma dynamics in various environments, such as cosmic phenomena, advanced material processing technologies, nuclear fusion reactor safety, and the enhancement of medical device sterilization (such as sterilization of surgical endoscopes).

Abstract Despite their importance to a variety of plasma processes in Jupiter's magnetosphere, properties of the thermal and suprathermal electrons have been sparsely reported in past literature. To address this gap, we used data from Juno 's Jovian Auroral Distributions Experiment Electron sensors to survey electron properties in Jupiter's magnetodisc between M‐shells of M = 9–40 and magnetic latitudes less than 20° within years 2016–2023. We fit the electron measurements with a sum of two anisotropic kappa distributions and provide estimates of the electron density, temperature, anisotropy, and thermodynamic kappa for the cold and hot populations, separately. We find that the electron temperature increases non‐adiabatically with M‐shell and the cold electron temperature anisotropy is most extreme near M = 15. We also provide estimates of the ambient electron properties near the Galilean moons, which should be treated with caution considering the many caveats of the fitting process, which is often associated with large uncertainties.

Abstract Auroral electron forcing with energy greater than tens keVs impacts D‐region ionization in high‐latitude regions, exhibiting increased energy from midnight to dawn. This increase, known as spectrum hardening, is characterized by a higher energy at the electron‐precipitation peak flux and/or a gradual power‐law spectrum gradient. The former relates to the Maxwellian component of the spectrum, while the latter pertains to the Kappa distribution component. However, limited research has distinguished these two components. This study investigated the dependence of the D‐region ionization on geomagnetic activity using electron density data from the European Incoherent Scatter (EISCAT) radar in Norway and cosmic noise absorption in Finland, sorted by the SuperMAG Auroral Electrojet index (SME). An inversion method derived differential energy spectra from the EISCAT‐measured height‐resolved electron density. Statistical spectrum analysis by fitting the Kappa distribution function revealed that spectrum hardening from midnight to dawn at auroral latitudes is mainly driven by the Kappa distribution component, characterized by a gradual power‐law spectrum gradient for moderately high geomagnetic activities (SME ≥300 nT). Conversely, for SME <300 nT, the Maxwellian component primarily also contributes to spectrum hardening. This study is the first to specify the energy range contributing to spectrum hardening.