The Fokker-Planck Collision Operator

The velocity-space diffusion and drag operator describing cumulative small-angle Coulomb collisions.

Why Fokker-Planck

In a plasma, particles interact through the long-range Coulomb force, so a given particle undergoes many simultaneous weak deflections rather than rare hard collisions. The cumulative effect of these small-angle encounters is described by the Fokker-Planck operator, which appears on the right side of the kinetic equation as a term modeling collisions: df/dt collision term = drag plus diffusion in velocity space.

Structure

The operator has two parts: a dynamical friction (drag) that slows fast particles toward the bulk, and a diffusion that spreads the distribution in velocity. Written with the Rosenbluth potentials, these coefficients are integrals over the field-particle distribution, making the operator nonlinear when a species collides with itself. The dominance of small-angle scattering appears through the Coulomb logarithm multiplying the whole operator, which is the accumulated factor by which many distant weak deflections outweigh the rare close encounter. Because the deflections are correlated with the field-particle distribution, the operator conserves particle number, momentum, and energy when integrated over the interacting species.

Consequences

The Fokker-Planck operator relaxes any distribution toward a Maxwellian, sets collision and slowing-down frequencies, and determines how fast ions from beams or fusion reactions thermalize. Its velocity dependence explains why fast particles are collisionless while thermal particles are collisional, and it underlies runaway-electron generation when the electric-field acceleration outruns the collisional drag on fast electrons.

Relevance

Fast-ion slowing-down, neutral-beam and radio-frequency heating deposition, alpha-particle thermalization, and runaway-electron risk during disruptions all require Fokker-Planck modeling. For the Hyperion D-T breeder concept, the slowing-down of 3.5 MeV fusion alphas onto the bulk plasma, which delivers self-heating, is a Fokker-Planck calculation in design-stage modeling of a machine not yet built.