The dynamics of a noninteracting colloidal fluid plunged in a Gaussian quenched-random external field is investigated based on a recently developed field-theoretic perturbation scheme. Starting with the Dean-Kawasaki equation for the microscopic density field and the associated Martin–Siggia–Rose–Janssen–de Dominicis dynamical action, its guiding principle is the requirement that the fluctuation–dissipation relation be explicitly preserved in the equilibrium dynamics.
At equilibrium, the first-order bare theory reproduces the mode-coupling theory (MCT) equations for the problem at hand, albeit with a memory integral written in terms of the correlation function of the pure system. It allows one to recover in a straightforward way many previous results on the long-time tails in fluids in disorder. After a renormalization step, a self-consistent theory is obtained which shows strong structural similarity with the MCT, but whose predictions somewhat improve upon those of the latter. Indeed, although a spurious ergodicity-breaking transition for the density fluctuations is still present, a normal diffusive behavior always holds at long time, in agreement with known rigorous results. Moreover, the theory is free from the spurious infrared singularities that impair the MCT at low space dimensions.
Out of equilibrium, the perturbation theory is applied in the steady state reached when all the particles are driven with the same constant external force. The resulting first-order bare theory is very close in structure to the MCT for active microrheology in colloidal suspensions. A rich analytic structure is unveiled, made of a variety of limiting singular behaviors and simple crossovers. Remarkable similarities with the behavior of the driven lattice Lorentz gas suggest a large degree of universality in the dynamics of driven fluids in disorder.