= Solution
The long-<wavelength> limit in <2/d> is \b[<resonant heating>]. Most electrons screen the slowly varying spatial field. The noise nevertheless excites collective <Langmuir waves>, whose sharp temporal spectrum lies near $\omega=\pm\omega_{pe}$. Electrons with $v\simeq\omega/k$ remain in phase with a wave and exchange <energy> with it. A stable tail absorbs net wave <energy> through <Landau damping>. The heating is initially deposited in narrow tail regions of <velocity space>, even though the volume-integrated rate equals that in <2/c>.
The short-<wavelength> limit in <2/c> is \b[<stochastic heating>] in the terminology intended here. With $\epsilon\simeq1$, the imposed <electric field> is essentially unscreened. Its temporally uncorrelated kicks cause a random walk of electron <velocity>, so $\langle(\Delta v)^2\rangle=2D_0\Delta t$. Electrons throughout the thermal population contribute to the resulting <kinetic energy> increase; no narrow collective wave peak selects a small tail population.
Both mechanisms are driven by <white noise>, and the formal condition $\omega=kv$ enters the stationary <quasilinear diffusion> calculation in both cases. The physical distinction is the sharp collective <resonance> in <2/d> versus the broad direct random forcing in <2/c>, rather than the mere presence of randomness. In collisionless theory this heating is an increase of the averaged <kinetic energy>; subsequent <collisions> can establish <thermal equilibrium>.
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