Observable Dependence of Fluctuation-Dissipation Relations and Effective Temperatures

We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud’s trap model. We incorporate an arbitrary observable $m$ and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of $m$. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature $T_{\mathrm{eff}}$ in this model. Consequences for the wider applicability of an FDT-derived $T_{\mathrm{eff}}$ are discussed.