  We investigate the effective diffusion of a tracer immersed in an active
particle bath consisting of self-propelled particles. Utilising the Dean's
method developed for the equilibrium bath and extending it to the
nonequilibrium situation, we derive a generalized Langevin equation (GLE) for
the tracer particle. The complex interactions between the tracer and bath
particles are shown as a memory kernel term and two colored noise terms. To
obtain the effective diffusivity of the tracer, we use path integral technique
to calculate all necessary correlation functions. Calculations show the
effective diffusion decreases with the persistent time of active force, and has
rich behavior with number density of bath particles, depending on different
activity. All theoretical results regarding the dependence of such diffusivity
on bath parameters have been confirmed by direct computer simulation.
