Influence of thermal fluctuations on active diffusion at large P\'{e}clet numbers

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.