  Wavelet Monte Carlo dynamics simulations are used to study the dynamics of
passive particles in the presence of microswimmers, taking account of the
often-omitted thermal motion alongside the hydrodynamic flows generated by the
swimmers. Although the P\'{e}clet numbers considered are large, we find the
thermal motion to have a significant effect on the dynamics of our passive
particles, and can be included as a decorrelation factor in the velocity
autocorrelation with a decay time proportional to the P\'{e}clet number.
Similar decorrelation factors come from swimmer rotations, e.g.~run and tumble
motion, and apply to both entrainment and far field loop contributions. These
decorrelation factors lead to active diffusivity having a weak apparent power
law close to $Pe^{0.2}$ for small tracer-like particles at P\'{e}clet numbers
appropriate for E. coli swimmers at room temperature. Meanwhile, the reduced
hydrodynamic response of large particles to nearby forces has a corresponding
reduction in active diffusivity in that regime. Together, they lead to a
non-monotonic dependence of active diffusivity on particle size that can shed
light on similar behaviour observed in experiments by Patteson et al.
