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 (tmuch less thant(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 nonDebye 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 to\omega\(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+(vertical bar t-t'vertical bar lt(0))(-alpha)]>T. The results are discussed and compared with experiments.