Exact Fluctuating Hydrodynamics of the Scaled Light-Heavy Model

We study the exact fluctuating hydrodynamics of the scaled Light-Heavy model (sLH), in which two species of particles (light and heavy) interact with a fluctuating surface. This model is similar in definition to the unscaled Light-Heavy model (uLH), except it uses rates scaled with the system size. The consequence, it turns out, is a phase diagram that differs from that of the unscaled model. We derive the fluctuating hydrodynamics for this model using an action formalism involving the construction of path integrals for the probability of different states that give the complete macroscopic picture starting from the microscopic one. This is then used to obtain the two-point steady-state (static) correlation functions between fluctuations in the two density fields in the homogeneous phase. We show that these theoretical results match well with microscopic simulations away from the critical line. We derive an exponentially decaying form for the two-point steady-state correlation function with a correlation length that diverges as the critical line is approached. Finally, we also compute the dynamic correlations in the homogeneous phase and use them to determine the relaxation dynamics as well as the dynamic exponents of the system.