Liouville Theorem

Taking into account the conservation of the particles, the phase space density is conserved as well:

\frac{d}{d t} F (x, v, t) = 0 = (\frac{\partial}{\partial t} + \vec{v} \cdot \nabla_{x} + \frac{d \vec{v}}{d t} \cdot \nabla_{v}) F (x, v, t)

Where $\frac{d \vec{v}}{d t}$ is given by the equation of motion for a particle under the action of electromagnetic fields

\frac{d}{d t} {\vec{v}}_{i} (t) = \frac{q}{m} [{\vec{E}}_{m} ({\vec{x}}_{i} (t), t) + {\vec{v}}_{i} (t) \times {\vec{B}}_{m} ({\vec{x}}_{i} (t), t)]

where the fields E and B are given by the microscopic Maxwell's equations. This holds true for collisionless plasma and in the absence of Diffusion and Scattering

Also we shouldn't have any sources or sinks -- #Question