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 + (δx/δt)δf/δx + (δv/δt)δf/δv

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

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/δx + (q/m)(E + vxB).(δf/δv)=0
where E and B are the electric and magnetic fields respectively.

See also the Fokker-Planck equation.

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