Vlasov–Poisson¶

\[ \partial_t f(t,\mathbf{x},\mathbf{v}) + \mathbf{v}\nabla_{\mathbf{x}} f(t,\mathbf{x},\mathbf{v}) + \frac{q}{m} \mathbf{E}(t,\mathbf{x}) \nabla_{\mathbf{v}} f(t,\mathbf{x},\mathbf{v}) = 0 \]

\[ -\Delta \phi(t,\mathbf{x}) = 1- \rho(t, \mathbf{x}) \]