  A direct connection between the magnitude of the violation of the
fluctuation-response relation (FRR) and the rate of energy dissipation is
presented in terms of field variables of nonequilibrium systems. Here, we
consider the density field of a colloidal suspension either in a relaxation
process or in a nonequilibrium steady state driven by an external field. Using
a path-integral representation of the temporal evolution of the density field,
we find an equality that relates the magnitude of the violation of the FRR for
scalar and vector potentials of the velocity field to the rate of energy
dissipation for the entire system. Our result demonstrates that the violation
of the FRR for field variables captures the entropic component of the
dissipated free energy.
