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

Abstract

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 $χ(t,t_w)$ versus the autocorrelation function is different from that of the kinetic Ising model. However, splitting $χ(t,t_w)$ into a stationary and an aging term $χ(t,t_w)=χ_{st}(t-t_w)+χ_{ag}(t,t_w)$, we find $χ_{ag}(t,t_w)\sim t_w^{-a_χ}f(t/t_w)$, and a numerical value of $a_χ$ consistent with $a_χ=1/4$, as in the kinetic Ising model.