The fluctuation-dissipation relation in an Ising model without detailed balance

We consider the modified Ising model introduced by de Oliveira et al. [J.Phys.A {\bf 26}, 2317 (1993)], where the temperature depends locally on the spin configuration and detailed balance and local equilibrium are not obeyed. We derive a relation between the linear response function and correlation functions which generalizes the fluctuation-dissipation theorem. In the stationary states of the model, which are the counterparts of the Ising equilibrium states, the fluctuation-dissipation theorem breaks down due to the lack of time reversal invariance. In the non-stationary phase ordering kinetics the parametric plot of the integrated response function $\chi (t,t_w)$ versus the autocorrelation function is different from that of the kinetic Ising model. However, splitting $\chi (t,t_w)$ into a stationary and an aging term $\chi (t,t_w)=\chi_{st}(t-t_w)+\chi_{ag}(t,t_w)$, we find $\chi_{ag}(t,t_w)\sim t_w^{-a_\chi}f(t/t_w)$, and a numerical value of $a_\chi$ consistent with $a_\chi =1/4$, as in the kinetic Ising model.