  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 $\omega^\ast$ 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 $AK = KA^{T}$ can be satisfied if and only if
$\omega^\ast = \omega_c$ . Our coarse-grained model calculations can be easily
generalized to other coupled systems.
