Renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

We study the field theoretic renormalized perturbation theory for a toy model of fluctuating nonlinear hydrodynamics (FNH) of compressible liquids. The toy model contains a density-like and a momentum-like variable without any spatial dependence. We present a detailed derivation of a set of coupled equations among correlation and response functions for these variables. In particular, we focus on how the static limit of the correlation and response functions can be achieved in the renormalized perturbation theory. Numerical methods of solving these equations at a given order of the loop expansion are explained and the results for the one-loop theory are given in detail. The simple nature of the toy model enables us to compare the static limit obtained from the exact solution with that of the one-loop order. This shows explicitly the range of validity of the one-loop theory in the field theoretic formulation.