Temporally correlated zero-range process with open boundaries: Steady state and fluctuations

We study an open-boundary version of the on-off zero-range process introduced in Hirschberg et al. [Phys. Rev. Lett. 103, 090602 (2009)]. This model includes temporal correlations which can promote the condensation of particles, a situation observed in real-world dynamics. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices, and also explore the large deviation properties of the particle current. Analytical and numerical calculations show that, although the particle distribution is well described by an effective Markovian solution, the probability of rare currents differs from the memoryless case. In particular, we find evidence for a memory-induced dynamical phase transition.

Figure 1

Non-Markovian ZRP on a one-dimensional lattice with open boundaries. Each site has a hidden variable $\tau$, whose values are represented by the positions of a gear, which controls the departure rate. When $\tau$ assumes value zero no departure is possible and the corresponding state is referred to as $\mathrm{OFF}$. This lockdown occurs in conjunction with the arrival of a particle.

Figure 2

Monte Carlo time evolution of the occupation profile of the on-off ZRP on a one-dimensional lattice. (a) Rates ${\mu }_{n,\tau }=n$ if $\tau >0,{\mu }_{n,\tau }=0$ otherwise, and $(\alpha ,\beta ,\gamma ,\delta ,p,q,c)=(0.1,0.2,0,0,1,0,0.05)$. Only correlations due to the blockade mechanism are present. The particles organize in traveling clusters. Their speed is mainly governed by $c$. (b) Same parameters as the former case, except $(\beta ,p)=({10}^4,{10}^4)$. The particles jump almost simultaneously to the next site as soon as the blockade is removed. Each particle cluster tends to occupy a single site. The drift proceeds with a rate $\simeq c$. (c) Same parameters as (a), except ${\mu }_{n,\tau }=1$ if $\tau >0$ and $n>0,{\mu }_{n,\tau }=0$ otherwise, and $c=0.15$. As a result of the attractive interparticle interaction, the clusters with more particles travel slower than the less populated ones. This mechanism enhances congestion.

Figure 3

Occupation probability distribution of the one-site system for constant $\left({\mu }_n=\mu ,n>0\right)$ and linear $\left({\mu }_n=n\right)$ microscopic departure rates. The arrival and departure rates are $\alpha =0.1$ and $\beta =0.2$, respectively.

Figure 4

Density profile $\langle n_l\rangle$, and variance profile ${{\sigma }_l}^2$ (insets), on a chain of length $L=20$ with (a) constant and (b) linear departure rates. The results obtained within the mean-field approximation (line vertices) are compared with the results computed by means of Monte Carlo simulations (markers). TA and PA refer to rates $(\alpha ,\beta ,\gamma ,\delta ,p,q)=(0.2,0.3,0,0,1,0)$ and $(\alpha ,\beta ,\gamma ,\delta ,p,q)=(0.1,0.2,0.1,0.1,0.55,0.45)$, respectively.

Figure 5

The mean-field probability distribution (line vertices) of the site occupation numbers in a chain of length $L=5$ are checked against simulations (markers). Symbols $+,\times ,\circ ,\square ,▵$ and solid, dotted, dashed, dotted-dashed, dotted-dotted-dashed lines of the corresponding color (grayscale) refer to sites $l=1,2,3,4,5$, respectively. Parameter combinations as in Fig. 4.

Figure 6

Simulation results for the cross correlation $C_{11,l}$ of on-off ZRP on a chain of length $L=20$ with ${\mu }_n=n$. (a) PA case. Adjacent sites have negatively correlated occupation numbers. Hopping-rate combinations as in Fig. 4. Inset, (b). TA case. Spatial cross correlations appear weaker than the PA case, but with longer range. As $c$ grows the correlation is gradually lost and a factorized solution is realistic.

Figure 7

Congestion transition for the on-off ZRP with $\mu =1$ on a open chain of length $L=20$. The parameter $\kappa$ obtained by Monte Carlo simulations is plotted against $c$ for a TA case and a PA case. Hopping-rate combinations as in Fig. 4. The mean-field congestion thresholds are $c_{\mathrm{mf}}^{\mathrm{TA}}=0.4$ and $c_{\mathrm{mf}}^{\mathrm{PA}}\simeq 1.4$, respectively. These values are pinpointed by the mean-field approximated local ${\kappa }_l$ (light lines) of the site $l$ where the congestion sets in first.

Figure 8

Real part of the spectrum of the finite-capacity $s$-modified Hamiltonian for parameters $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.2,0.3,0,0,0.5)$ and ${\mu }_n=n$. For $s<s_1$ the smaller eigenvalue converges to $A_0=\alpha (1-e^{-s})$, while for $s>s_1$ it converges to $c$.

Figure 9

SCGF of the on-off ZRP with ${\mu }_n=n$ for (a) TA hopping rates $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.2,0.3,0,0,0.1)$ and (b) PA hopping rates $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.1,0.2,0.1,0.1,0.1)$. Points are data from the cloning algorithm $\mathcal{N}={10}^4,t={10}^4$. The SCGF is systematically overestimated for small values of $s$. We expect a better approximation but a slow convergence for larger ensembles and longer simulation times.

Figure 10

Rate function $\widehat{e}\left(j\right)$ for the on-off TA process with $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.2,0.3,0,0,0.01)$ and ${\mu }_n=n$ (solid line). The points are simulation data for the finite-time rate function $-ln\left[\mathrm{Prob}\right(J/t=j\left)\right]/t$ computed at $t=200,300,400,500,600,700$ (top to bottom).

Figure 11

Phase diagram, based on Eq. (58), for the current fluctuations in the PA process with ${\mu }_n=1$ and $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.1,0.2,0.1,0.1.0.5)$. The lines $s_1$ and $s_4$ correspond to first-order dynamical phase transitions while $s_2$ and $s_3$ mark second-order transitions.

Figure 12

SCGF of the on-off ZRP with $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.1,0.2,0.1,0.1,0.5)$ and ${\mu }_n=1$. Points are data from the cloning simulations $\mathcal{N}={10}^4,t={10}^4$. Dotted line is the SCGF of the Markovian ZRP $\left(c\to \infty \right)$ with same boundary rates. Solid line is the analytic approximation (62, 63, 64). The SCGF of the ZRP with $s\text{-independent}$ departure rate $w_c$ would overlap the solid line at this scale.

Figure 13

Rate function for the on-off ZRP with $(\alpha ,\beta ,\gamma ,\delta ,c)=(0.1,0,2,0.1,0.1,0.5)$ and ${\mu }_n=1$. Points are data for $-ln\left[\text{Prob}\right(J/t=j\left)\right]/t$ from standard Monte Carlo simulation at times $t=100,200,300,400,500,1000,2000$ (top to bottom). The solid line is the analytical approximation for $t\to \infty$.

Figure 14

Simulation results for the SCGF in a five-site on-off ZRP with ${\mu }_n=1$ and $(\alpha ,\beta ,\gamma ,\delta ,p,q,c)=(0.1,0.2,0.1,0.1,0.55,0.45,0.5)$. The solid line is the $c\text{-independent}$ expression for the lowest eigenvalue of the $s$-modified Hamiltonian for the five-site Markovian ZRP [38].