Non-equilibrium statistical mechanics tool for the study of space plasma: The Ehrenfest procedure in Earth’s radiation belts and Superstatistics in a magnetized plasma

An intriguing issue in plasma physics , approached from the standpoint of non-equilibrium Statistical Mechanics, is the derivation of properties for collisionless plasmas using distinct differential equations for probability density. These equations , such as the Vasov equation and the Fokker-Planck equation, offer crucial insights into out-of-equilibrium systems and have been applied in interpreting electromagnetic phenomena. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions using a test particle simulation, showing good agreement between theory and numerical results for these observables. We conclude that this procedure provides a powerful tool for the study of dynamical systems and statistical mechanics out of equilibrium, and opens perspectives for applications in other systems with probabilistic continuity.

For another hand it is widely acknowledged that the statistical properties of space plasmas often deviate from canonical distributions, such as Maxwell-Boltzmann or Jüttner, which are applicable only to systems in equilibrium. Observational data spanning several decades have revealed the ubiquity of nonthermal 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, sparking discussions about the validity of nonextensive 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. Indeed, we know that the velocity distribution of a particle in a non-collisional plasma in steady state must follow the Superstatistics formalism. 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. This approach aims to provide new perspectives for understanding and modeling phenomena in space plasmas and temperature in systems out of equilibrium.