Partial differential equation describing the evolution of the distribution function of a collisionless plasma.

The distribution function *f* describes the state of a fluid in terms of the position **x** and velocity **v** of every constituent particle at every time *t*. (i.e. f=f(**x**,**v**,t))

Differentiating f with respect to time

df/dt= δf/δt + (δ**x**/δt)δf/δ**x** + (δ**v**/δt)δf/δ**v**

In the absence of collisions the total differential df/dt is equal to zero. Note that

**v**=δ**x**/δt

**F**/m=δ**v**/δt

where

**F** and

*m* are the

force and

mass respectively. Furthermore, in a

plasma the force in question is the

electromagnetic force and thus the Vlasov equation may be written

δf/δt + **v**.δf/δ**x** + (q/m)(**E** + **v**x**B**).(δf/δ**v**)=0

where

**E** and

**B** are the

electric and

magnetic fields respectively.

See also the Fokker-Planck equation.