Weak Pinning and Long-Range Anticorrelated Motion of Phase Boundaries in Driven Diffusive Systems

We show that domain walls separating coexisting extremal current phases in driven diffusive systems exhibit complex stochastic dynamics with a subdiffusive temporal growth of position fluctuations due to long-range anticorrelated current fluctuations and a weak pinning at long times. This weak pinning manifests itself in a saturated width of the domain wall position fluctuations that increases sublinearly with the system size. As a function of time $t$ and system size $L$, the width $w(t,L)$ has a scaling behavior $w(t,L)=L^{3/4}f(t/L^{9/4})$, with $f(u)$ constant for $u\gg 1$ and $f(u)\sim u^{1/3}$ for $u\ll 1$. An Orstein-Uhlenbeck process with long-range anticorrelated noise is shown to capture this scaling behavior. The exponent $9/4$ is a new dynamical exponent for relaxation processes in driven diffusive systems.

Figure 1

(a) Illustration of the open TASEP with repulsive nearest-neighbor interaction $V$: Particles are injected with rate ${\mathrm{\Gamma }}_L(n_2)$ from the left particle reservoir, ejected with rate ${\mathrm{\Gamma }}_R(n_{L-1})$ to the right reservoir, and perform unidirectional jumps with rates ${\mathrm{\Gamma }}_i(n_{i-1},n_{i+2})$ (3) inside the system. (b) Current-density relation for $V=2V_c=4\ln (3)$ in the closed TASEP. (c) Nonequilibrium phase diagram following from (b) by applying the extremal current principle (1). In the yellow-blue shaded square the two maximum current phases II and VII coexist. The red square, circle, and line in this area mark reservoir densities, for which DW dynamics were simulated.

Figure 2

Simulated density profiles in the coexistence region $\mathrm{II}+\mathrm{VII}$ at randomly selected times (thin solid lines) for (a) $({\rho }_L,{\rho }_R)=(0.8,0.3016)$ and $L=500$, (b) $({\rho }_L,{\rho }_R)=(0.8,0.2)$ and $L=500$, and (c) $({\rho }_L,{\rho }_R)=(0.8,0.2)$ and $L=4000$. The thick blue lines mark the long-time averaged densities. The dashed vertical lines in (a) indicate the displacement $⟨x⟩$ of the mean DW position from the center. The dash-dotted vertical lines in (b) and (c) indicate the standard deviation $\pm w_{\mathrm{sat}}/L$ of the saturated DW position fluctuations around the center. The insets show two short-time averages of the fluctuating particle density (noisy green and olive curves), as well as histograms of the distribution of ghost particle positions. Their mean value marks the microscopic position of the DW (dashed vertical lines).

Figure 3

Mean DW position $⟨x⟩$ as a function of $L$ for fixed ${\rho }_L=0.8$ and various ${\rho }_R$ corresponding to points on the red dashed line shown in the coexisting region $\mathrm{II}+\mathrm{VII}$ in Fig. 1. For large $L$, $⟨x⟩\to 0$, demonstrating that the displacement $⟨x⟩/L$ in Fig. 2 is a finite size effect.

Figure 4

(a) Width $w(t,L)$ of DW position fluctuations [see Eq. (5)] as a function of time for various system sizes $L$. (b) Scaled width $w(t,L)/L^{3/4}$ as a function of scaled time $t/L^{9/4}$ demonstrating the scaling law (2). The legend in (b) applies to (a) also. Solid lines indicate the results from the Ornstein-Uhlenbeck process given by Eq. (8) with parameters ${\kappa }_0=1.133$ and $C_{\infty }=0.0225$.