  By direct application of the fluctuation-dissipation theorem, we numerically
calculate the fundamental dimensional fluctuations of crystalline CaF2
whispering gallery resonators in the case of structural damping, and the limit
that this noise imposes on the frequency stability of such resonators at both
room and cryogenic temperatures. We analyze elasto-optic noise - the effect of
Brownian dimensional fluctuation on frequency via the strain-dependence of the
refractive index - a noise term that has so far not been considered for
whispering-gallery resonators. We find that dimensional fluctuation sets a
lower limit of 1E-16 to the Allan deviation for a 10-millimeter-radius sphere
at 5 K, predominantly via induced fluctuation of the refractive index.
