The Fokker-Planck Equation
The kinetic equation that adds small-angle Coulomb collisions as a drift-and-diffusion operator in velocity space.
Collisions as diffusion
In a plasma, most collisions are grazing: many tiny deflections accumulate rather than one large scattering event. The Fokker-Planck equation captures this by adding a velocity-space operator to the Vlasov equation, written as a drag term plus a diffusion term:
df/dt + v . grad_x f + (q/m)(E + v x B).grad_v f = C[f]
C[f] = -grad_v.(A f) + (1/2) grad_v grad_v : (D f)A is the dynamical friction (slowing-down) vector and D the velocity-space diffusion tensor. Both are built from the Rosenbluth potentials, integrals over the background distribution that encode the cumulative effect of many small-angle Coulomb encounters.
Slowing down and thermalization
The friction term drives fast particles toward the bulk; the diffusion term spreads them out. Together they relax any distribution toward a Maxwellian at the collision rate set by the Coulomb logarithm. This is exactly how fusion-born alpha particles or neutral-beam ions deposit their energy.
How it is solved numerically
- Finite-difference or finite-element discretization in (v_parallel, v_perp) with the collision operator treated implicitly for stability
- Monte-Carlo collision operators that apply random kicks reproducing the friction and diffusion coefficients
- Linearized operators that preserve conservation of particles, momentum, and energy exactly
The nonlinearity through the Rosenbluth potentials is often handled by lagging them or solving them as auxiliary Poisson-like equations each step.
Where it matters
Fast-ion slowing down, current drive by injected beams, runaway-electron generation, and alpha-particle heating in a D-T plasma are all Fokker-Planck problems. For the Hyperion breeder, modeling how fusion alphas thermalize and heat the bulk is a Fokker-Planck calculation.