On relaxation processes in a completely ionized plasma

Keywords: plasma, generalized Lorentz model, relaxation coefficients, collision integral operator, spectral theory, one-polynomial approximations, functional hypothesis


Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma.  The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of  Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.


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Author Biographies

Alexander I. Sokolovsky, Oles Honchar Dnipro National University; Dnipro, Ukraine

Department of Theoretical Physics


Sergey A. Sokolovsky, Prydniprovska State Academy of Civil Engineering and Architecture; Dnipro, Ukraine

Sokolovsky Sergey Alexandrovych

Department of Physics, Associate Professor

Oleh A. Hrinishyn, Oles Honchar Dnipro National University; Dnipro, Ukraine

Department of Theoretical Physics

PhD Sudent



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How to Cite
Sokolovsky, A., Sokolovsky, S., & Hrinishyn, O. (2020). On relaxation processes in a completely ionized plasma. East European Journal of Physics, (3), 19-30. https://doi.org/10.26565/2312-4334-2020-3-03