Effective temperature in nonequilibrium steady states of Langevin systems with a tilted periodic potential

We theoretically study Langevin systems with a tilted periodic potential. It is known that the ratio Theta of the diffusion constant D to the differential mobility mu(d) 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 Theta 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 Theta plays the role of the temperature in the large scale description of the system and that Theta can be determined directly in experiments, without measurements of the diffusion constant and the differential mobility. Hence the relation D= mu(d) Theta among the independent measurable quantities D, mu(d), and Theta can be interpreted as an extension of the Einstein relation.