Vlasov–Maxwell
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Vlasov–Maxwell¶
Equations 1d2v:
\[\begin{split}
\begin{aligned}
&&\partial_t f(t,x,\mathbf{v}) + v_1 \partial_{x_1} f(t,x,\mathbf{v}) + \mathbf{E}(t, x) \cdot \nabla_{\mathbf{v}} f(t, x, \mathbf{v}) + B v_2 \partial_{v_1} f(t,x,\mathbf{v}) - B v_1 \partial_{v_2} f(t,x,\mathbf{v}) = 0, \\
&& \partial_t B(t,x) = - \partial_{x_1} E_2, \\
&& \partial_t E_2 = - \partial_{x_1} B - \int_{\mathsf{R}^1} v_2 f(t, x, \mathbf{v}) \, \mathrm{d}v,\\
&& \partial_t E_1 = - \int_{\mathsf{R}^1} v_1 f(t, x, \mathbf{v}) \, \mathrm{d}v.\end{aligned}
\end{split}\]