Brownian motion in an aging medium.
The motion of a particle in an aging medium can be described by the generalized Langevin equation, in the limit of long waiting time t(w) where the medium is in a quasistationary regime at the scale of the observation times investigated (t<< t(w) ) . In this framework, we analyze the link between the Brownian motion and the effective temperature which characterizes the out-of-equilibrium properties of the medium. This effective temperature involves a frequency-dependent effective temperature T(eff) (omega) formally identical to a generalized susceptibility. The analytical results are reported in the case when T(eff) (omega) is mapped to the universal non-Debye power-law ac response met for instance in dielectrics. In the particular case where the viscous friction coefficient is a power law gamma (omega) proportional, variant mid R:omegamid R:(delta-1) , contact is made with the heuristic expression T(eff) =T [1+ (omega/ omega(0) )(alpha) ] , postulated in prior experimental and theoretical works. A closed analytic form of the time correlation function of the medium coordinate (the noise force) C(FF) (t- t(') ) = F (t) F ( t(') ) is obtained, in the subdiffusive regime (delta<1) where C(FF) (t- t(') ) is a regular function. This time correlation is long range. We also determine another effective temperature T(')(eff) (t- t(') ) of the medium, usually defined in aging systems as the temperature associated with the violation of the fluctuation-dissipation theorem in its time formulation. This temperature takes the form T(')(eff) (t- t(') ) =T [1+ ( mid R:t- t(') mid R: / t(0) )(-alpha) ] >T . The results are discussed and compared with experiments.