Dynamics of a noninteracting colloidal fluid in a quenched Gaussian random potential: A time-reversal-symmetry-preserving field-theoretic approach

We develop a field-theoretic perturbation method preserving the fluctuation-dissipation relation (FDR) for the dynamics of the density fluctuations of a noninteracting colloidal gas plunged in a quenched Gaussian random field. It is based on an expansion about the Brownian noninteracting gas and can be considered and justified as a low-disorder or high-temperature expansion. The first-order bare theory yields the same memory integral as the mode-coupling theory (MCT) developed for (ideal) fluids in random environments, apart from the bare nature of the correlation functions involved. It predicts an ergodic dynamical behavior for the relaxation of the density fluctuations, in which the memory kernels and correlation functions develop long-time algebraic tails. A FDR-consistent renormalized theory is also constructed from the bare theory. It is shown to display a dynamic ergodic-nonergodic transition similar to the one predicted by the MCT at the level of the density fluctuations, but, at variance with the MCT, the transition does not fully carry over to the self-diffusion, which always reaches normal diffusive behavior at long time, in agreement with known rigorous results.