Fundamentals of Plasma Physics 2025, 41 - Fokker-Planck Equation

Consider a particle with velocity v(t), subject to random collisions, influencing it. Define F(v, Δv) as the conditional probability that a particle with v at time t, at will at time t + Δt have velocity v + Δv. The sum of all conditional probabilities must equal to 1. Define f(v, t) = ∫f(v-Δv, t-Δt)F(v-Δv, Δv)dΔv to sum up all possible ways to obtain a given velocity. Substitution identifies f(v, t), which then defines the Fokker-Planck equation by derivation. Define, too, the deflection angle θ

Averaging is done by calculating the collision effects in Δt, where the functional dependence of Δv on the scattering angle θ is determined, and then used to calculate the weighted average Δv for all possible impact parameters and all possible azimuthal angles of incidence. An averaging is performed over all possible field particle velocities weighted according to their probability, weighted according to the field particle distribution function f_F(v_F). The difference between relative velocities is

Define the mean velocity u via nu = ∫vfdv, where n = ∫fdv. With

If the test particle is much faster than ion and electron thermal velocities, then ξ_{i, e} >> 1. If it's much slower than ion and electron thermal velocities, then ξ_{i, e} << data-preserve-html-node="true" 1. If the particle is much faster than ion thermal velocity, but much slower than electron thermal velocity, then ξ_i >> 1, ξ_e << data-preserve-html-node="true" 1. Write normal slowing down time through the relation &partial;_t = -u/τ_s. Each case views this quantity differently.

An external electrical field will accelerate ions and electrons in opposite directions, introducing frequent collisions and, as a consequence, collective frictional drag. The dynamics of these can be approached with the typical Maxwellian distribution functions. The ion thermal velocity is much smaller than the electron thermal velocity on account of the mass difference, so the ion distribution function is much narrower than the electron distribution function, and can be considered mono-energetic. The net force of the resulting beam impacting the electrons can be translated into a frictional drag, acting against the electric field's acceleration.

The inclusion of the error function assumes that the relative drift between electron and ion mean velocities u_rel is much smaller than electron thermal velocity. It implies that -(x^-1erf x)' ≈ 4x/e√π, which has a max of 0.43 for x = 0.9. The electron temperature in natural units sees the balance between accelerations to be impossible above a critical Dreicer electric field strength.