  The regulation of electron heat transport in high-beta, weakly collisional,
magnetized plasma is investigated. A temperature gradient oriented along a mean
magnetic field can induce a kinetic heat-flux-driven whistler instability
(HWI), which back-reacts on the transport by scattering electrons and impeding
their flow. Previous analytical and numerical studies have shown that the heat
flux for the saturated HWI scales as the inverse of electron beta. These
numerical studies, however, had limited scale separation and consequently large
fluctuation amplitudes, which calls into question their relevance at
astrophysical scales. To this end, we perform a series of particle-in-cell
simulations of the HWI across a range of electron beta and temperature-gradient
length scales under two different physical setups. The saturated heat flux in
all of our simulations follows the expected inverse-electron-beta scaling,
supporting the robustness of the result. We also use our simulation results to
develop and implement several methods to construct an effective collision
operator for whistler turbulence. The results point to an issue with the
standard quasi-linear explanation of HWI saturation, which is analogous to the
well-known 90-degree scattering problem in the cosmic ray community. Despite
this limitation, the methods developed here can serve as a blueprint for future
work seeking to characterize the effective collisionality caused by kinetic
instabilities.
