Time integration

\[ \partial_t f(t, x, v) + v_x \partial_x f + v_y \partial_y f + \frac{E_x}{m} \partial_{v_x} f + \frac{E_y}{m} \partial_{v_x} f = 0 -\Delta \phi(x) = \sum_s \rho_{q,s}(x,y) \rho_{q,s}(x, y) = \int q_s f_s(x,y,v_x,v_y) dv_x v_y E = - \nabla \phi \]