Brownian motion in an aging medium

Abstract

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 ${\mathcal{T}}_{\mathrm{eff}}\left(\omega \right)$ formally identical to a generalized susceptibility. The analytical results are reported in the case when ${\mathcal{T}}_{\mathrm{eff}}\left(\omega \right)$ 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 $\gamma \left(\omega \right)\propto {\mid \omega \mid }^{\delta -1}$, contact is made with the heuristic expression $T_{\mathrm{eff}}=T[1+{(\omega ∕{\omega }_0)}^{\alpha }]$, postulated in prior experimental and theoretical works. A closed analytic form of the time correlation function of the medium coordinate (the noise force) $C_{FF}(t-t^{\prime })=⟨F\left(t\right)F\left(t^{\prime }\right)⟩$ is obtained, in the subdiffusive regime $(\delta <1)$ where $C_{FF}(t-t^{\prime })$ is a regular function. This time correlation is long range. We also determine another effective temperature $T_{\mathrm{eff}}^{\prime }(t-t^{\prime })$ 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 $T_{\mathrm{eff}}^{\prime }(t-t^{\prime })=T[1+{(\mid t-t^{\prime }\mid ∕t_0)}^{-\alpha }]>T$. The results are discussed and compared with experiments.