The Maxwell-Boltzmann approximation for ion kinetic modeling

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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

\displaystyle n_e =e^{q_{e} \phi / \theta }

in which {n_e, q_e, \theta,\phi} 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 {\widetilde{f} _{+}(x,v,t)} and {\widetilde{f}_{-}(x,w,t)} be the corresponding density distribution functions for ions and electrons, respectively; here, {(v,w)} represent particle velocity variables for ions and electrons belonging to { \mathbb{R}^{d}} (here {d=2} or {3}), and {x} denotes the space variable belonging to a periodic torus or an open set of {\mathbb{R}^{d}} with a boundary, and {t} is the time. In absence of magnetic fields, the dynamics of the plasma is modeled by the following well-known system

\displaystyle \partial _{t}\widetilde{f}_{-}+w\cdot \nabla _{x}\widetilde{f}_{-}-\frac{q_{e} }{m_{e}}\widetilde{E}\cdot \nabla _{w}\widetilde{f}_{-} = \widetilde{Q}_{-}( \widetilde{f}_{-})

\displaystyle \partial _{t}\widetilde{f}_{+}+v\cdot \nabla _{x}\widetilde{f}_{+}+\frac{q_{e} }{m_{i}}\widetilde{E}\cdot \nabla _{v}\widetilde{f}_{+} = 0

where {m_{e},m_{i}} denote the electron and ion mass, {q_{e}} the elementary charge (for the sake of simplicity we assume that the ion charge is equal to {1}). The electrostatic field is given by {\widetilde{E}=-\nabla _{x}\widetilde{\phi }} solving the usual Poisson equation, and the operator {\widetilde{Q}_{-}( \widetilde{f}_{-})} 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 {\widetilde{\theta }} and a density given by the well-known Maxwell-Boltzmann relation is used for electrons density.

If we introduce non-dimensional parameter

\displaystyle \varepsilon =\sqrt{\frac{m_{e}}{m_{i}}} \rightarrow 0

and look at the scaled distribution functions

\displaystyle f_{-}(v)=\frac{1}{\epsilon^3 N_{ref}}\widetilde{f}_{-}(v/\varepsilon ),\qquad f_{+}(v)=\frac{1}{N_{ref}}\widetilde{f}_{+}(v).

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

\displaystyle \varepsilon \partial _{t}f_{-}+v\cdot \nabla _{x}f_{-}+\nabla _{x}\phi \cdot \nabla _{v}f_{-} = \eta _{\epsilon }Q(f_{-})

\displaystyle \partial _{t}f_{+}+v\cdot \nabla _{x}f_{+}-\nabla _{x}\phi \cdot \nabla _{v}f_{+} =0

and the Poisson equation for the electric potential {\phi } reads as

\displaystyle -\lambda _{D}^{2}\Delta _{x}\phi =\left\langle f_{+}\right\rangle -\left\langle f_{-}\right\rangle .

Here, {\lambda _{D}=\sqrt{\epsilon ^{0}\theta _{ref}/(q_{e}^{2}N_{ref})} } 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:

\displaystyle \qquad \lim_{\varepsilon \rightarrow 0}\eta _{\varepsilon }\varepsilon ^{-1}=\infty ,\qquad \lim_{\varepsilon \rightarrow 0}\eta _{\varepsilon } < +\infty.

As {\epsilon \rightarrow 0}, we obtain in the limit the electron density distribution

\displaystyle {f_-}(x,v,t)=n_e (x,t)\Big (\frac{\beta(t) }{2\pi}\Big)^{\frac{d}2} e^{-\beta(t) \frac{ |v|^2}2}, \qquad n_e(x,t) = e^{\beta(t) \phi(x,t)}

with {\beta(t)} being the inverse of the electron temperature. The Vlasov equation for ions remains the same, whereas the Poisson equation now reads

\displaystyle -\lambda _{D}^{2}\Delta \phi + e^{\beta \phi}= \left\langle f_{+}\right\rangle

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 {\frac{|v|^2}{2} - \phi }. 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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