Month: April 2025

The Vlasov theory for relativistic plasmas near vacuum

ttn12 Kinetic theory, Plasma Physics , , , ,

In plasma physics, the dynamics of collisionless charged particles can often be modeled by the relativistic Vlasov-Maxwell system, namely the transport equation

\displaystyle \partial_t f + \hat v \cdot \nabla_x f + \mathcal K\cdot \nabla_v f = 0 \ \ \ \ \ (1)

for the unknown one-particle density distribution {f(t,x,v)} in the phase space {\mathbb{R}_x^3 \times \mathbb{R}^3_v}, coupled with the Lorentz force {\mathcal K = E + \hat v \times B}, whose electromagnetic fields are computed though the classical Maxwell equations

\displaystyle \begin{aligned} \partial_t B + \nabla_x \times E = 0, \qquad \nabla_x \cdot E &= \rho[f], \\ - \partial_t E + \nabla_x \times B = {\bf j}[f], \qquad \nabla_x \cdot B & =0. \end{aligned} \ \ \ \ \ (2)

The particles travel with relativistic velocities {\hat v = v/\langle v\rangle} with {\langle v\rangle =\sqrt{1+|v|^2}} (normalizing the speed of light to be one), and the charge and current densities are defined by {\rho[f] = \int f(t,x,v) dv - n_{\mathrm{ion}}} and {{\bf j}[f] = \int \hat v f(t,x,v)\; dv - {\bf j}_{\mathrm{ion}}} for some fixed constant background densities {n_{\mathrm{ion}}} and {{\bf j}_{\mathrm{ion}}} (i.e. for sake of presentation, we simply focus on the dynamics of excited electrons, fixing a uniform ion background).

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