Einstein relation and effective temperature for systems with quenched disorder

According to the Einstein relation the ratio between independent measurements of fluctuations (e.g., diffusivity) and the response to a weak external field (e.g., mobility) is equal to the thermal temperature when the system is kept close to thermal equilibrium. For strongly disordered systems, which are not self-averaging this ratio is a random variable and hence in this sense the Einstein relation is not valid. Thus effective temperatures found using the fluctuation dissipation ratio are at least in some cases stochastic variables. This scenario is tested with the quenched trap model. An average over an ensemble of systems yields an averaged effective temperature, which is compared with results obtained from the mean-field continuous-time random-walk model.