  We investigate a class of nonequilibrium media described by Langevin dynamics
that satisfies the local detailed balance. For the effective dynamics of a
probe immersed in the medium, we derive an inequality that bounds the violation
of the second fluctuation-dissipation relation (FDR). We also discuss the
validity of the effective dynamics. In particular, we show that the effective
dynamics obtained from nonequilibrium linear response theory is consistent with
that obtained from a singular perturbation method. As an example of these
results, we propose a simple model for a nonequilibrium medium in which the
particles are subjected to potentials that switch stochastically. For this
model, we show that the second FDR is recovered in the fast switching limit,
although the particles are out of equilibrium.
