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

We carry 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 $\sigma$, and 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 relation $(L∕2\pi )c\chi =\sigma \Phi$, where $c$ represents the propagation velocity of density fluctuations.

Figure 1

$\gamma \left(t\right)$ is plotted with error bars in the case $\overline{\rho }=0.5$, $L=64$, and $\mid {\widehat{V}}_1\mid ∕\sqrt{L}=0.1$ [see Eq. (10)]. The slope of the fitted line (the dotted line) in the early stage $(t<150\mathrm{MCS})$ corresponds to $D{(2\pi ∕L)}^2$. The slope of $\gamma \left(t\right)$ in the late stage $(t>150\mathrm{MCS})$ differs from $D{(2\pi ∕L)}^2$.

Figure 2

$D$ as a function of $L$ for the case $\overline{\rho }=19∕64$ (solid circles) and $\overline{\rho }=0.5$ (open circles). The dotted line and the solid line represent $0.042\sqrt{L}+0.20$ and $0.068\sqrt{L}+0.0413$, respectively.

Figure 3

The profiles of ${⟨{\eta }_x⟩}_{E,V}^{\mathrm{s}}$ under the influence of $V_x=\Delta \sin (2\pi x∕L)$ in the case $\overline{\rho }=19∕64$ and $L=64$. $\Delta =0.1$, 0.2, 0.3, and 0.4 correspond to the circle, square, triangle, and plus, respectively.

Figure 4

The profiles of ${⟨{\eta }_x⟩}_{E,V}^{\mathrm{s}}$ under the influence of $V_x=\Delta \sin (2\pi x∕L)$ in the case $\overline{\rho }=0.5$ and $L=64$. $\Delta =0.1$, 0.2, 0.3, and 0.4 correspond to the circle, square, triangle, and plus, respectively.

Figure 5

$\Theta$ and $\Phi$ as functions of $L$ in the cases $\overline{\rho }=19∕64$ (solid symbols) and $\overline{\rho }=0.5$ (open symbols). $\Theta \left(L\right)$ and $\Phi \left(L\right)$ correspond to the circle and square, respectively. The solid line, the dotted line, and the dash-dotted line represent $0.36\sqrt{L}+1.4$, $0.21L$, and $0.58\sqrt{L}+0.26$. Note that $T=2$.

Figure 6

$D\chi$ as a function of $\sigma \Theta$. The cases $\overline{\rho }=19∕64$ and $\overline{\rho }=0.5$ correspond to the solid symbols and the open symbols, respectively. The cases $L=64$, 128, and 256 correspond to the square, triangle, and circle, respectively.

Figure 7

$R\left(t\right)$ as a function of $C\left(t\right)$ in the case $\overline{\rho }=0.5$ and $L=64$ during an interval $0⩽t⩽800\mathrm{MCS}$. The slope of the thin line represents ${\Theta }^{-1}$.

Figure 8

$B\left(t\right)t$ as a function of $t$ in the case $\overline{\rho }=0.5$ and $L=128$. The guided line (the dashed line) represents $B\left(t\right)t\propto t^{4∕3}$.

Figure 9

$k^{-1}c\chi$ as a function of $\sigma \Phi$, where $k=2\pi ∕L$, for the case $\overline{\rho }=19∕64$. Note that for the case $\overline{\rho }=0.5$, $c=\Phi =0$. The cases $L=64$, 128, 192, and 256 correspond to the square, triangle, diamond, and circle, respectively. The straight line represents $(L∕2\pi )c\chi =\sigma \Phi$. The error bars represent the standard deviations of the statistical samples. In addition to the statistical error bars, there exists a fitting error which is less than 5% of the quantities.

Figure 10

In the case $H_0\left(\eta \right)=0$ (ASEP), $\gamma \left(t\right)$ is plotted with error bars in the case $\overline{\rho }=0.5$, $L=64$, and $\mid {\widehat{V}}_1\mid ∕\sqrt{L}=0.1$ [see Eq. (10)]. The slope of the fitted line (thin line) in the early stage $(t<150\mathrm{MCS})$ corresponds to $D{(2\pi ∕L)}^2$.