Partial differential equation describing the evolution of the distribution function of a collisional plasma (or other stochastic system)..
The distribution function f=f(x,v,t) describes the particles of a gas by their locations and velocity at any instant. See the the writeup of the Vlasov equation which deals with the collisionless case. The effects of collisions can be added simply by including a term on the right hand side of that equation.
δf/δt + v.δf/δx + (q/m)(E + vxB).(δf/δv)= (δf/δt)col
where
E and
B are the
electric and
magnetic fields repectively. (δf/δt)
col is the change in the distribution function due to collsions. This
collisional term can be approximated in various ways.
In a plasma, the long-range Coulomb force between the constituent charged particles means that they rarely directly collide. Instead, when one particle approaches another it is deflected. In general, these deflections are small. A form of the collisional term has been derived based on these assumptions.
In plasma physics, the Fokker-Planck equation is of great importance and is used to
- study collisional transport.
- describe the slowing down of a beam of particles injected into the plasma.
- calculate characteristic relaxation times of test particles.
Source: 'Tokamaks' by John Wesson
Reference: The Fokker-Planck equation for a plasma and the Rosenbluth potentials are derived in the paper, Rosenbluth, M.N., MacDonald, W.M., and Judd, D.L 'Fokker-Planck equation for an inverse-square force', Physical Review 107, 1, (1957).