Long-range steady-state density profiles induced by localized drive

We show that the presence of a localized drive in an otherwise diffusive system results in steady-state density and current profiles that decay algebraically to their global average value, away from the drive in two or higher dimensions. An analogy to an electrostatic problem is established, whereby the density profile induced by a driving bond maps onto the electrostatic potential due to an electric dipole located along the bond. The dipole strength is proportional to the drive, and is determined self-consistently by solving the electrostatic problem. The profile resulting from a localized configuration of more than one driving bond can be straightforwardly determined by the superposition principle of electrostatics. This picture is shown to hold even in the presence of exclusion interaction between particles.

Figure 1

(a) Bond configuration with a single driving bond $(0,0)\to (1,0)$ and hopping rate $e=(1-\epsilon )$. Everywhere else the rate is 1. The $\oplus$ and $\ominus$ symbols denote the sign of the electric charge. The symbol $\odot$ represents the magnetic flux coming out of the plane, and $\otimes$ represents the flux going in. (b) A configuration of closed loop of driving bonds generating no electric charges. The magnetic flux through the central plaquette is four times larger than that of the other four neighboring plaquettes.

Figure 2

The algebraic decay of the density profile away from the driving bond in the positive $x$ direction.