  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.
