Density nonlinearities in field theories for a toy model of fluctuating nonlinear hydrodynamics of supercooled liquids

TL;DR

This paper investigates a simplified zero-dimensional model of fluctuating nonlinear hydrodynamics to understand density nonlinearities' effects on correlation functions and ergodic-nonergodic transitions in supercooled liquids.

Contribution

It compares two field-theoretic formulations of the toy model, revealing that one supports a transition while the other does not, highlighting the impact of nonlinearities on glassy dynamics.

Findings

DM-type theory does not support a sharp transition.

Alternative formulation admits an ergodic-nonergodic transition.

Numerical solutions for coupled dynamical equations are developed.

Abstract

We study a zero-dimensional version of the fluctuating nonlinear hydrodynamics (FNH) of supercooled liquids originally investigated by Das and Mazenko (DM) [Phys. Rev. A {\bf 34}, 2265 (1986)]. The time-dependent density-like and momentum-like variables are introduced with no spatial degrees of freedom in this toy model. The structure of nonlinearities takes the similar form to the original FNH, which allows one to study in a simpler setting the issues raised recently regarding the field theoretical approaches to glass forming liquids. We study the effects of density nonlinearities on the time evolution of correlation and response functions by developing field theoretic formulations in two different ways: first by following the original prescription of DM and then by constructing a dynamical action which possesses a linear time reversal symmetry as proposed recently. We show explicitly that, at the one-loop order of the perturbation theory, the DM-type field theory does not support a sharp ergodic-nonergodic transition, while the other admits one. The simple nature of the toy model in the DM formulation allows us to develop numerical solutions to a complete set of coupled dynamical equations for the correlation and response functions at the one-loop order.