In this short paper with C. Bardos, F. Golse, and R. Sentis, we are aimed to justify the Maxwell-Boltzmann approximation from kinetic models, widely used in the literature for electrons density distribution; namely, the relation

in which denote the electrons density, the elementary charge, the electron temperature, and the electric potential.

More precisely, we consider a plasma consisting of electrons and one kind of ions, which are charged particles moving in an electromagnetic field. Let and be the corresponding density distribution functions for ions and electrons, respectively; here, represent particle velocity variables for ions and electrons belonging to (here or ), and denotes the space variable belonging to a periodic torus or an open set of with a boundary, and is the time. In absence of magnetic fields, the dynamics of the plasma is modeled by the following well-known system

where denote the electron and ion mass, the elementary charge (for the sake of simplicity we assume that the ion charge is equal to ). The electrostatic field is given by solving the usual Poisson equation, and the operator accounts for the collisions between the electrons (for example, a binary Boltzmann or Fokker-Planck operator).

Such a model has been widely used in plasma physics. Since the electron/ion mass ratio is small, the characteristic time scale of the dynamics of ions is significantly larger than that of electrons. As a consequence, if one addresses a model for the ion dynamics, it is very classical to use a fluid modeling for the electrons, assuming they have reached the thermal equilibrium; that is to say, the distribution function is a Maxwellian function with an electron temperature and a density given by the well-known *Maxwell-Boltzmann relation* is used for electrons density.

If we introduce non-dimensional parameter

and look at the scaled distribution functions

with being the characteristic electron density, we end up with the non-dimensional Vlasov-Poisson-Boltzmann system:

and the Poisson equation for the electric potential reads as

Here, denotes the Debye length. In this paper, under certain regularity assumption, we prove that the Maxwell-Boltzmann relation for electrons, keeping the dynamics of ions at the kinetic level, is obtained under the scaling assumption:

As , we obtain in the limit the electron density distribution

with being the inverse of the electron temperature. The Vlasov equation for ions remains the same, whereas the Poisson equation now reads

which is often referred to as the Poisson-Poincare equation.

The relaxation to the equilibrium of the form of a Maxwellian is precisely due to the presence of the collision operators, without which the equilibrium is of the form of a function of the particle energy . In this paper, we also prove that the reduced ions problem (that is, the Vlasov equation for ions, coupled with the Poisson-Poincare equation and the energy conservation) is wellposed globally in time.

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