Extended Einstein relations with a complex effective temperature in a one dimensional driven lattice gas

We carried out numerical experiments on a one-dimensional driven lattice gas to elucidate the statistical properties of steady states far from equilibrium. By measuring the bulk density diffusion constant $D$, the conductivity $\sigma$, the intensity of density fluctuations $\chi$, we confirm that the Einstein relation $D\chi=\sigma T$, which is valid in the linear response regime about equilibrium, does not hold in such steady states. Here, $T$ is the environment temperature, and the Boltzmann constant is set to unity. Recalling that the Einstein relation provided the first step in the construction of linear response theory, we attempt to extend it to a generalized form valid in steady states far from equilibrium. In order to obtain new relations among measurable quantities, we define a complex effective temperature $\Theta-i\Phi$ from studying the static response of the system to a slowly varying potential in space. Replacing $T$ in the Einstein relation by the real part of the effective temperature $\Theta$, we numerically confirm that the relation $D\chi=\sigma\Theta$ holds in the nonequilibrium steady states far from equilibrium that we study. In addition to this extended form, we find the new relation $(L/2\pi)c\chi=\sigma\Phi$, where $c$ represents the propagation velocity of density fluctuations.