  Fuzzy dark matter of ultra-light axions has gained attention, largely in
light of the galactic scale problems associated with cold dark matter. But the
large de Broglie wavelength, believed to possibly alleviate these problems,
also leads to fluctuations that place constraints on ultra-light axions. We
adapt and extend a method, previously devised to describe the effect of gaseous
fluctuations on cold dark matter cusps, in order to determine the imprints of
ultra-light axion haloes on the motion of classical test particles. We first
evaluate the effect of fluctuations in a statistically homogeneous medium of
classical particles, then in a similar system of ultra light axions. In the
first case, one recovers the classical two body relaxation time (and diffusion
coefficients) from white noise density fluctuations. In the second situation,
the fluctuations are not born of discreteness noise but from the finite de
Broglie wavelength; correlation therefore exists over this scale, while white
noise is retained on larger scales, elucidating the correspondence with
classical relaxation. The resulting density power spectra and correlation
functions are compared with those inferred from numerical simulations, and the
relaxation time arising from the associated potential fluctuations is
evaluated. We then apply our results to estimate the heating of disks embedded
in axion dark haloes. We find that this implies an axion mass $m \ga 2 \times
10^{-22} {\rm eV}$. We finally apply our model to the case of the central
cluster of Eridanus II, confirming that far stronger constraints on $m$ may in
principle be obtained, and discussing the limitations associated with the
assumptions leading to these.
