Stochastic heating and entropy cascade in a solvable model of collisionless plasma turbulence

A (1+1)-D nearly collisionless plasma stirred by a stochastic external turbulent electric field, analogous to the Kraichnan model of passive advection, is considered. The mean effect on the particle distribution function is turbulent diffusion in velocity space—so-called stochastic heating. Associated with this heating is the generation of fine structure in the distribution function, which can be characterized by the collisionless (Casimir) invariant C₂ = ∫ dx dv δf²—a proxy for minus the entropy of the perturbed distribution function. C₂ is found to be transferred from large to small scales in position and velocity space via a phase-space `entropy' cascade driven by particle streaming and nonlinear field-particle interactions. The flux of C₂ in wavenumber space is dominated by the ‘critical balance’ region in phase space where the linear and nonlinear time scales are comparable. Integrating over velocity wavenumbers, the k-space flux of C₂ is constant down until a ‘Kolmogorov’ length scale that tends to zero as the collision frequency does. These results reveal that stochastic heating is in fact an entropy-cascade process and that phase mixing can be suppressed in the inertial range of a turbulent collisionless plasma while simultaneously being an effective means of dissipation.