One-dimensional transport of interacting particles: Currents, density profiles, phase diagrams, and symmetries

Driven lattice gases serve as canonical models for investigating collective transport phenomena and properties of nonequilibrium steady states. Here we study one-dimensional transport with nearest-neighbor interactions both in closed bulk systems and in open channels coupled to two particle reservoirs at the ends of the channel. For the widely employed Glauber rates we derive an exact current-density relation in the bulk for unidirectional hopping. An approach based on time-dependent density functional theory provides a good description of the kinetics. For open systems, the system-reservoir couplings are shown to have a striking influence on boundary-induced phase diagrams. The role of particle-hole symmetry is discussed, and its consequence for the topology of the phase diagrams. It is furthermore demonstrated that systems with weak bias can be mapped onto systems with unidirectional hopping.

Figure 1

Sketch of particle jump with rate $\Gamma (n_{i-1},n_{i+2})$ from site $i$ to site $(i+1)$ for the different possibilities of occupations of sites $(i-1)$ and $(i+2)$.

Figure 2

Bulk current-density relation $j\left(\rho \right)$ for various interaction strengths $V$.

Figure 3

Couplings of the system to the (a) left and (b) right reservoirs mediated by the $\alpha$ and $\beta$ rates. Indices 0 and 1 refer to the occupation of the sites next to the target site (for $\alpha$ rates) and to the initial site (for $\beta$ rates). The primed rates are for jumps from and to boundary sites of the system.

Figure 4

Boundary-induced phase diagrams of the NESS at $V=2V_{☆}$ for (a) the bulk-adapted and (b) the equilibrated-bath couplings. Thick solid and thick dashed lines mark first- and second-order phase transitions for the TASEP. Corresponding thin lines are for the ASEP with $F=2$. In (b) the symbols refer to kinetic Monte Carlo (KMC) results and the lines to MCAK results.

Figure 5

Time evolution of ${\rho }_i$ for (a) bulk-adapted and (b) equilibrated-bath couplings of the systems to the reservoirs for ${\rho }_{\mathrm{L}}=0.9$, ${\rho }_{\mathrm{R}}=0.6$, $V=2V_{☆}$, $N=1000$, and ${\rho }_i(t=0)=0$. KMC results are marked by symbols and MCAK results by lines. In the KMC simulations, averages were performed over ${10}^6$ different configurations. The inset in (b) zooms out the oscillatory behavior at the boundaries.

Figure 6

(a)–(c) Current-density relations $j\left(\rho \right)$ for the ASEP at various $V$ and three different bias strengths $F$. In (d) the current-bias relation is shown for $\rho =0.3$. Symbols refer to KMC data, lines to MCAK results, and the assignment of lines and symbols to the different interaction strengths $V$ is given in the legend of (a).

Figure 7

KMC results for the joint probabilities $p\left(1100\right)$ and $p\left(0011\right)$ of finding a configuration $\{n_{i-1}=1,n_i=1,n_{i+1}=0,n_{i+2}=0\}$ and its reversed configuration $\{n_{i-1}=0,n_i=0,n_{i+1}=1,n_{i+2}=1\}$ in the bulk part of the NESS for the ASEP defined in Eq. (24). The interaction strength is $V=2V_{☆}$ and the particle concentration $\rho =0.5$.