New hyperbolic statistics for the equilibrium distribution function of interacting electrons


  • Yu.A. Zelenin International Foundation «Chronograph», Ukraine
  • T.A. Bilyi Subbotin Institute of Geophysics of the National Academy of Sciences of Ukraine, Ukraine



cold plasma, electron distribution function, elliptical functions, atmospheric electrons


New statistics of a low-parameter distribution of the sech (ε, µ) type are presented, which reproduce the results of plasma simulation by the method of dynamics of many particles (DMP) with high accuracy. The distribution is based on a conceptual model of a two-component plasma — virtual quasiparticles of negative energy (exciton phase ε<0); the scattering region of positive energy (gas phase ε>0). Optimization and elementary estimates of the applicability of the sech (ε, µ) distribution statistics were made after the results of DMP experiments. The sech (ε,µ) distribution reduces the number of parameters of the three-piece DMP distribution from 4 energy diffusion coefficients (D1, D2, D3, D4) to two — the chemical potential µ and the asymmetry coefficient α. The functional relationship D1, D2, D3, D4 with the chemical potential of the system µ in the sech (ε, µ) distribution is introduced in a similar way to the Einstein relation between mobility and energy diffusion constants. The functional variety of the differential equation belongs to the family of elliptic functions. It is much wider than the hyperbolic solution given, which has significant physical application for complex values of the energy ε. The proposed simplified scheme grounded in the physical interpretation of negative energies can be written for the electrometric electrons of the atmosphere, which previously presented significant methodological difficulties. The chemical potentials of the fluid (metastable states) and gas phases are presented as functions of the plasma imperfection parameter. The problem is posed as an application to the problem of electrometric electrons in the atmosphere. The proposed distribution is not represented in mathematical statistics and statistical physics; it is new and extremely relevant.


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How to Cite

Zelenin, Y. ., & Bilyi, T. . (2023). New hyperbolic statistics for the equilibrium distribution function of interacting electrons. Geofizicheskiy Zhurnal, 44(6), 112–119.