  We study the effective diffusion constant of a Brownian particle linearly
coupled to a thermally fluctuating scalar field. We use a path integral method
to compute the effective diffusion coefficient perturbatively to lowest order
in the coupling constant. This method can be applied to cases where the field
is affected by the particle (an active tracer), and cases where the tracer is
passive. Our results are applicable to a wide range of physical problems, from
a protein diffusing in a membrane to the dispersion of a passive tracer in a
random potential. In the case of passive diffusion in a scalar field, we show
that the coupling to the field can, in some cases, speed up the diffusion
corresponding to a form of stochastic resonance. Our results on passive
diffusion are also confirmed via a perturbative calculation of the probability
density function of the particle in a Fokker-Planck formulation of the problem.
Numerical simulations on simplified systems corroborate our results.
