## Vlasov equation

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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
 (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
 (2)
which is known as Vlasov equation.
References:
• [1]^ A. A. Vlasov (1938). "On Vibration Properties of Electron Gas" (in Russian). J. Exp. Theor. Phys. 8 (3): 291.