Studying a Non-Collisional Plasma with Superstatistics considerations.
Authors: Tamburrini Abiam , Moya Pablo , Davis Sergio 
Published: March 2024

It is widely acknowledged that the statistical properties of space plasmas often deviate from canonical distributions, such as Maxwell-Boltzmann or J&#252;ttner, which are applicable only to systems in equilibrium. Observational data spanning several decades have revealed the ubiquity of non-thermal particle distributions in space plasmas near Earth. Various phenomenological empirical distributions, including the kappa suprathermal and non-thermal Cairns distributions, have been introduced to address this phenomenology. These distributions represent suprathermal deviations from Maxwellian equilibrium and are expected to exist in systems with low pressure and density conditions, such as the solar wind, where collisional events are negligible, and the assumption of thermal equilibrium is challenging to justify. Q-exponential distributions, derived from the generalization of entropy developed by Tsallis, are not exempt from inconsistencies[1], sparking discussions about the validity of non-extensive statistical mechanics in different contexts. One of the challenges lies in the application of fundamental concepts such as temperature, traditionally defined in the context of thermodynamic equilibrium. Since most plasmas near Earth are in a steady state but not in thermodynamic equilibrium, the premise of thermal equilibrium and associated concepts, such as temperature, becomes untenable.
Given that finding a steady state in a system out of equilibrium is highly non-trivial, superstatistics seek to describe these states using a few parameters and the concept of hyper-ensembles[2]. Indeed, we know that the velocity distribution of a particle in a non-collisional plasma in steady state must follow the Superstatistics formalism[3]. Therefore, considering the origin of empirical distributions in modeling space plasma phenomena is not a settled issue. In this work, we will present a deeper analysis from the perspective of superstatistics for the description of space plasmas. We start from a linear approximation of the Vlasov Equation and apply superstatistics considerations to explore its scope and possible interpretation in dispersion relations for a magnetized plasma, extending previous analysis on electrostatic plasma waves[4]. This approach aims to provide new perspectives for understanding and modeling phenomena in space plasmas and temperature in systems out of equilibrium.
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[1] S. Press&#233;, &#8220;Nonadditive entropy maximization is inconsistent with bayesian updating,&#8221;Phys. Rev. E, vol. 90, p. 052149, Nov 2014.
[2] C. Beck and E. Cohen, &#8220;Superstatistics,&#8221; Physica A: Statistical Mechanics and its Applica-tions, vol. 322, pp. 267&#8211;275, 2003
[3] S. Davis, G. Avaria, B. Bora, J. Jain, J. Moreno, C. Pavez, and L. Soto, &#8220;Single-particlevelocity distributions of collisionless, steady-state plasmas must follow superstatistics,&#8221;Phys. Rev. E, vol. 100, p. 023205, Aug 2019.
[4] K. Ourabah, &#8220;Demystifying the success of empirical distributions in space plasmas,&#8221;Phys. Rev. Research, vol. 2, p. 023121, May 2020.
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