The Vlasov equation [1] is a differential equation describing the dynamical evolution of the phase space distribution function.


The time evolution of the phase space distribution function $f(x,p,t)$ is governed by Liouville's theorem which asserts that in case of dissipationless and collisionless dynamics, the phase space distribution function is constant along the trajectories of the system

$$\frac {df}{dt} = \frac {\partial f}{\partial t} + \frac {\partial x}{\partial t} \frac {\partial f}{\partial x} + \frac {\partial p}{\partial t} \frac {\partial f}{\partial p} = 0.$$

(1)

Using the definition of the momentum $p=m\dot {x}$ and Newton's law $\dot {p} = F = -\nabla U$ for a force $F$ derived from a potential $U$, this can be rewritten as

$$\frac {\partial f}{\partial t} + \frac {p}{m} \frac {\partial f}{\partial x} - \frac {\partial U}{\partial x} \frac {\partial f}{\partial p} = 0,$$

(2)

which is known as Vlasov equation.