  The effect of electrostatic microturbulence on fast particles rapidly
decreases at high energy, but can be significant at moderate energy. Previous
studies found that, in addition to changes in the energetic particle density,
this results in nontrivial changes to the equilibrium velocity distribution.
These effects have implications for plasma heating and the stability of
Alfv\'en eigenmodes, but make multiscale simulations much more difficult
without further approximations. Here, several related analytic model
distribution functions are derived from first principles with reasonable
approximations. A single dimensionless parameter characterizes the relative
strength of turbulence relative to collisions, and this parameter appears as an
exponent in the model distribution functions. Even the most simple of these
models reproduces key features of the numerical phase-space transport solution
and provides a useful a priori heuristic for determining how strong the effect
of turbulence is on the redistribution of energetic particles in toroidal
plasmas.
