  We consider a nearly collisionless plasma consisting of a species of `test
particles' in 1D-1V, stirred by an externally imposed stochastic electric
field. The mean effect on the particle distribution function is stochastic
heating. Accompanying this heating is the generation of fine-scale structure in
the distribution function, which we characterize with the collisionless
(Casimir) invariant $C_2 \propto \iint dx dv \, \langle f^2 \rangle$. We find
that $C_2$ is transferred from large scales to small scales in both position
and velocity space via a phase-space cascade enabled by both particle streaming
and nonlinear interactions between particles and the stochastic electric field.
We compute the steady-state fluxes and spectrum of $C_2$ in Fourier space, with
$k$ and $s$ denoting spatial and velocity wavenumbers, respectively. Whereas
even the linear phase mixing alone would lead to a constant flux of $C_2$ to
high $s$ (towards the collisional dissipation range) at every $k$, the
nonlinearity accelerates this cascade by intertwining velocity and position
space so that the flux of $C_2$ is to both high $k$ and high $s$
simultaneously. Integrating over velocity (spatial) wavenumbers, the $k$-space
($s$-space) flux of $C_2$ is constant down to a dissipation length (velocity)
scale that tends to zero as the collision frequency does, even though the rate
of collisional dissipation remains finite. The resulting spectrum in the
inertial range is a self-similar function in the $(k,s)$ plane, with power-law
asymptotics at large $k$ and $s$. We argue that stochastic heating is made
irreversible by this entropy cascade and that, while collisional dissipation
accessed via phase mixing occurs only at small spatial scales rather than at
every scale as it would in a linear system, the cascade makes phase mixing even
more effective overall in the nonlinear regime than in the linear one.
