Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale $\omega$ between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value $\omega ={\omega }_c$ when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ${\omega }_c$ in terms of the coupling parameters of the system. We show that ${\omega }_c$ is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ${\omega }_c$ from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ${\omega }^{*}$ of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix $A$ and compressibility matrix $K$. Our exact calculations show that Onsager-type current symmetry relation $AK=K{A}^T$ can be satisfied if and only if ${\omega }^{*}={\omega }_c$. Our coarse-grained model calculations can be easily generalized to other coupled systems.