A fluctuation-dissipation relationship-preserving field theory for interacting Brownian particles: one-loop theory and mode coupling theory

We develop a field theoretical treatment of a model of interacting Brownian particles. We pay particular attention to the requirement of time reversal invariance and the fluctuation-dissipation relationship (FDR). The method used is a modified version of the auxiliary field method due originally to Andreanov, Biroli and Lefevre (2006 J. Stat. Mech. P07008). We recover the correct diffusion law when the interaction is dropped as well as the standard mode coupling equation in the one-loop order calculation for interacting Brownian particle systems. It is noteworthy that despite our starting dynamical model containing the bare inter-particle interaction potential, it cancels out and the direct correlation function emerges in the end result due to our use of the correct static input.