  A novel equality relating the rate of energy dissipation to a degree of
violation of the fluctuation-response relation (FRR) in nonequilibrium Langevin
systems is described. The FRR is a relation between the correlation function of
the fluctuations and the response function of macroscopic variables. Although
it has been established that the FRR holds in equilibrium, physical
significance of violation of the FRR in nonequilibrium systems has been under
debate. Recently, the authors have found that an extent of the FRR violation is
related in a simple equality to the rate of energy dissipation into the
environment in nonequilibrium Langevin systems. In this paper, we fully explain
the FRR, the FRR violation, and the new equality with regard to a Langevin
model termed a Brownian motor model, which is considered as a simple model of a
biological molecular motor. Furthermore, applications of our result to
experimental studies of molecular motors are discussed, and, as an
illustration, we predict the value of a new time constant regarding the motion
of a KIF1A, which is a kind of single-headed kinesin.
