= Solution
For a regular thermal $f_0$ and noise at $k\lambda_{\mathrm{De}}\gg1$, the susceptibility $\epsilon-1$ is small at thermal <phase velocities>: its scale is $\omega_{pe}^2/(k^2v_{\mathrm{the}}^2)$. To leading order the electrons cannot appreciably screen such short-scale fields, and <2/b> becomes independent of <velocity>:
$$
D(v)\simeq D_0=\frac{e^2}{m_e^2}\sum_k k^2A_k.
$$
The electron <kinetic energy> per unit volume is $\mathcal K=(m_e/2)\int v^2f_0\,dv$. Assume $v^2D f_0'$ and $vf_0$ vanish at infinity. One <integration by parts> gives the general heating formula
$$
\frac{d\mathcal K}{dt}=-m_e\int_{-\infty}^{\infty}vD(v)f_0'(v)\,dv.
$$
For constant $D_0$, a second <integration by parts> and $\int f_0\,dv=n_e$ give
$$
\boxed{\frac{d\mathcal K}{dt}=m_en_eD_0=\frac{e^2n_e}{m_e}\sum_k k^2A_k.}
$$
This 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>.
Back to article page
