Stochastic heating

Stochastic heating increases random particle kinetic energy through decorrelating electromagnetic kicks. A simple one-dimensional model makes the normalization explicit. Let $f(v,t)$ be a velocity distribution with number density $n=\int fdv$, and suppose constant-coefficient velocity-space diffusion givesThe coefficient has dimensions of squared velocity per time, and a freely diffusing component has variance $2Dt$. For sufficiently decaying tails, two applications of integration by parts yieldThe first velocity moment is unchanged, so this is growth of random motion rather than acceleration of the mean flow. In three independent isotropic components with coefficient $D$ in each, the heating rate per volume is instead $3mnD$. More generally each diagonal component contributes its own diffusion coefficient.

This idealization describes externally supplied, rapidly decorrelating forcing; its energy budget is not that of an isolated particle population. Screening, resonant self-consistent fields, collisions and velocity-dependent diffusion can change the result. In other plasma settings, stochastic heating also describes irregular orbits driven by spatial structure or overlapping resonances; that mechanism must be specified separately.