Long-range correlations in a locally driven exclusion process

We show that the presence of a driven bond in an otherwise diffusive lattice gas with simple exclusion interaction results in long-range density-density correlation in its stationary state. In dimensions $d>1$ we show that in the thermodynamic limit this correlation decays as $C(\mathbf{r},\mathbf{s})\sim {(r^2+s^2)}^{-d}$ at large distances $r$ and $s$ away from the drive with $|\mathbf{r}-\mathbf{s}|\gg 1$. This is derived using an electrostatic analogy whereby $C(\mathbf{r},\mathbf{s})$ is expressed as the potential due to a configuration of electrostatic charges distributed in $2d$ dimension. At bulk density $\rho =1/2$ we show that the potential is that of a localized quadrupolar charge. At other densities the same is correct in leading order in the strength of the drive and it is argued numerically to be valid at higher orders.

Figure 1

Exclusion process on a periodic square lattice with asymmetric jump rates between sites $-{\mathbf{e}}_1$ and $\mathbf{0}$, and symmetric elsewhere.

Figure 2

The distribution of charges on the $(r_1{\mathbf{e}}_1,s_1{\mathbf{e}}_1)$ plane. The charge ($\text{a}$) ${\sigma }_1$, ($\text{b}$) ${\sigma }_2$, and ($\text{c}$) ${\sigma }_3$.

Figure 3

Contour plot on the $(r_1{\mathbf{e}}_1,s_1{\mathbf{e}}_1)$ plane of the numerical solution of the Poisson equation (5) on a four-dimensional lattice.

Figure 4

Log-log plot of the $c(\mathbf{r},\mathbf{s})$ along three different directions. The dashed lines denote a power-law fit with $1/x^4$. The ($▴$) data are shifted by one unit in the logarithmic scale for better presentation.