 Solution

For a regular thermal $f_0$ and noise at $k{\lambda }_{\mathrm{De}}\gg 1$, the susceptibility $ϵ-1$ is small at thermal phase velocities: its scale is ${\omega }_{pe}^2/(k^2{v}_{\mathrm{the}}^2)$. To leading order the electrons cannot appreciably screen such short-scale fields, and b becomes independent of velocity:The electron kinetic energy per unit volume is $\mathcal{K}=(m_e/2)\int v^2{f}_{0}dv$. Assume $v^{2}D{f}_0^{\prime }$ and $v{f}_0$ vanish at infinity. One integration by parts gives the general heating formulaFor constant $D_0$, a second integration by parts and $\int f_{0}dv=n_e$ giveThis is the leading short-wavelength result. With vanishing boundary flux, the diffusion equation also conserves particle number; constant $D_0$ conserves the mean momentum while increasing the random kinetic energy.