Fluctuation-dissipation in active matter

In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle's response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = k_BT/ζ, where D is the particle's self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and k_BT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium-they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry-and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy.