We theoretically study Langevin systems with a tilted periodic potential. It is known that the ratio [Formula presented] of the diffusion constant [Formula presented] to the differential mobility [Formula presented] is not equal to the temperature of the environment (multiplied by the Boltzmann constant), except in the linear response regime, where the fluctuation dissipation theorem holds. In order to elucidate the physical meaning of [Formula presented] far from equilibrium, we analyze a modulated system with a slowly varying potential. We derive a large scale description of the probability density for the modulated system by use of a perturbation method. The expressions we obtain show that [Formula presented] plays the role of the temperature in the large scale description of the system and that [Formula presented] can be determined directly in experiments, without measurements of the diffusion constant and the differential mobility. Hence the relation [Formula presented] among the independent measurable quantities [Formula presented], [Formula presented], and [Formula presented] can be interpreted as an extension of the Einstein relation. © 2004 The American Physical Society.