Nonequilibrium temperatures in steady-state systems with conserved energy

We study a class of nonequilibrium lattice models describing local redistributions of a globally conserved quantity, which is interpreted as an energy. A particular subclass can be solved exactly, allowing us to define a statistical temperature $T_{\mathrm{th}}$ along the same lines as in the equilibrium microcanonical ensemble. We compute the response function and find that when the fluctuation-dissipation relation is linear, the slope $T_{\mathrm{FD}}^{-1}$ of this relation differs from the inverse temperature $T_{\mathrm{th}}^{-1}$. We argue that $T_{\mathrm{th}}$ is physically more relevant than $T_{\mathrm{FD}}$, since in the steady-state regime, it takes equal values in two subsystems of a large isolated system. Finally, a numerical renormalization group procedure suggests that all models within the class behave similarly at a coarse-grained level, leading to a parameter that describes the deviation from equilibrium. Quantitative predictions concerning this parameter are obtained within a mean-field framework.