Exclusion Processes with Internal States

We introduce driven exclusion processes with internal states that serve as generic transport models in various contexts, ranging from molecular or vehicular traffic on parallel lanes to spintronics. The ensuing nonequilibrium steady states are controllable by boundary as well as bulk rates. A striking polarization phenomenon accompanied by domain wall motion and delocalization is discovered within a mesoscopic scaling. We quantify this observation within an analytic description providing exact phase diagrams. Our results are confirmed by stochastic simulations.

Figure 1

Illustration of an exclusion model with two internal states. Particles in states $\uparrow$ ($\downarrow$) enter with rates ${\alpha }^{\uparrow }$ (${\alpha }^{\downarrow }$), move unidirectionally to the right within the lattice, may flip at rate $\omega$, and leave the system at rates ${\beta }^{\uparrow }$ (${\beta }^{\downarrow }$), always respecting Pauli’s exclusion principle.

Figure 2

The polarization phenomenon appearing in the IN region (${\alpha }^{\uparrow }=0.4$, ${\alpha }^{\downarrow }=0.2$, ${\beta }^{\uparrow }=0.2$, ${\beta }^{\downarrow }=0.45$, and $\Omega =0.5$). A domain wall forms in the density profile of spin-up states, while spin-down stays in a LD phase. The spin currents, shown in the inset, are both continuous. Solid lines correspond to the analytical solution, while dashed lines indicate results from stochastic simulations on lattices with $L=2000$ (blue or dark gray) and $L=10000$ (red or light gray).

Figure 3

Phase diagram for equal injection rates ${\alpha }^{\uparrow }={\alpha }^{\downarrow }\equiv \alpha$. Phases for spin-up and -down are shown in panels (a) and (b), respectively, as a function of $\alpha$ and ${\beta }^{\uparrow }$ for fixed values ${\beta }^{\downarrow }=0.3$ and $\Omega =0.15$. Phase separation (shaded areas) arises in the IN as well as the EX region. The delocalization-transition line (thick line) is identical in (a) and (b). Thin lines correspond to continuous transitions.