Generalized Einstein relation for aging processes

Physical aging appears in many systems ranging from glassy/granular materials, blinking quantum dots to laser-cooled atoms. Aging is a process with three fingerprints: (i) slow, non-exponential relaxation, (ii) breaking of time-translation-invariance, and (iii) dynamical scaling. Here, we show that all these features are present in our minimal Langevin model for aging. A natural extension of the Einstein relation, which was expected to be true in an equilibrium state, is conjectured to hold in aging processes where both the damping and the temperature decrease with time in power-law forms. The generalized Einstein relation can be used to tackle the difficult problem of determining non-ergodic behaviours. The model shows a power-law-type diffusion away from the critical point and a logarithmic Sinai-type ultra-slow diffusion at the critical point. Application to granular gases is also discussed. The authors propose a minimal Langevin model with time-dependent noise, diffusion coefficient, and friction coefficient, which is appropriate to describe cooling environments (granular gases, laser cooling). Assuming that the temperature and the friction coefficient decay in a power-law manner, the generalized Einstein relation is analysed.