Stationary uphill currents in locally perturbed zero-range processes

Uphill currents are observed when mass diffuses in the direction of the density gradient. We study this phenomenon in stationary conditions in the framework of locally perturbed one-dimensional zero range processes (ZRPs). We show that the onset of currents flowing from the reservoir with smaller density to the one with larger density can be caused by a local asymmetry in the hopping rates on a single site at the center of the lattice. For fixed injection rates at the boundaries, we prove that a suitable tuning of the asymmetry in the bulk may induce uphill diffusion at arbitrarily large, finite volumes. We also deduce heuristically the hydrodynamic behavior of the model and connect the local asymmetry characterizing the ZRP dynamics to a matching condition relevant for the macroscopic problem.

Figure 1

Schematic representation of the ZRP-OB model (left panel) and the ZRP-CC model (right panel). Rates associated with the black sites are $pu$ towards the left and $qu$ towards the right.

Figure 2

Fugacity profile, for $R$ large, in the case $\alpha >\delta$ (left panel) and $\alpha =\delta$ (right panel).

Figure 3

Fugacity profile, for $R$ large, in the case $\alpha <\delta$ for $\epsilon <{\epsilon }_{\mathrm{c}}$ (left panel), $\epsilon ={\epsilon }_{\mathrm{c}}$ (central panel), and ${\epsilon }_{\mathrm{c}}<\epsilon$ (right panel).

Figure 4

Monte Carlo results for the almost everywhere symmetric ZRP-OB with $R=50$, $\epsilon =0.4$, $\alpha =0.2$, $\delta =0.3$. The initial datum used in the simulations is a uniform configuration with two particles per site. Squares and circles refer, respectively, to the independent particle and the simple exclusion-like models. The dashed lines represent the exact solution. Left panel: the total number of particles in the system is measured vs time. The open symbols are the instantaneous values, and the solid symbols are the time-averaged values. Time is measured from the beginning of the dynamics. The inset in the upper right corner is a magnification of the larger figure at short times and shows the rapid relaxation of the time-averaged total number of particles in the independent particle case to the theoretical value given in (25). Right panel: particle currents measured at the boundaries vs time. Open and solid symbols refer, respectively, to the left and the right boundary. Time is counted starting from the “thermalization” time $2\times {10}^6$, taken as the origin of the horizontal axis.

Figure 5

Monte Carlo results for the density profile of the almost everywhere symmetric ZRP-OB with $R=50$, $\epsilon =0.4$, $\alpha =0.2$, and $\delta =0.3$. Squares and circles refer, respectively, to the independent particle (left panel) and the simple exclusion-like (right panel) models. The dashed lines represent the exact solution given in (19). The stationary density profiles have been computed by averaging, after the thermalization time is reached, over a set of ${10}^6$ instantaneous particle configurations.

Figure 6

Results for the almost everywhere symmetric ZRP-BB model with $R=50$, $N=206$, $\lambda =0.25$, $\epsilon =0.05$ (circles), 0.2 (diamonds), 0.4 (squares). Left panel: stationary density profiles, obtained as discussed in Fig. 5. Open symbols denote the Monte Carlo measures and the dashed lines the exact solution in (37) and (38). Note that the average occupation numbers of the finite reservoirs in $x=0$ and in $x=2R+2$ have been multiplied by the factor $\lambda$ to fit within the figure. Right panel: particle currents measured on the bonds $(2R+1)-(2R+2)$ (open symbols) and $(2R+2)-\left(0\right)$ (solid symbols). The dashed lines represent the theoretical prediction in (41). Time, on the horizontal axis, is counted from the “thermalization” time $2\times {10}^6$, as in Fig. 4.

Figure 7

Comparison between the exact solution of the hydrodynamic problem (49, 50, 51) and the Monte Carlo measure of the time-dependent density profiles, averaged over a set of different realizations of the stochastic process. The parameters of the simulation are $\alpha =0.5$, $\delta =1$, $\epsilon =0.4$, and $R=50$. The profiles are plotted at times $t=0.001$ (triangles), $t=0.01$ (squares), $t=0.1$ (diamonds), and $t=0.5$ (circles). The gray and the black solid lines denote, respectively, the initial condition and the exact solution at the corresponding times. The profiles in the regions $[0,1/2]$ and $[1/2,1]$ are displayed in two separate panels to optimize the resolution of the plots.

Figure 8

Comparison between the exact solution of the hydrodynamic problem (49, 50, 51) and the Monte Carlo measure of the time-dependent density profiles, with $\alpha =\delta =0.5$, $\epsilon =0.4$, and $R=50$. Symbols are the same as those shown in Fig. 7.

Figure 9

Comparison between the exact solution of the hydrodynamic problem (49, 50, 51) (black solid lines) and the Monte Carlo measure, with $\alpha =0.5$, $\delta =1.0$, $\epsilon =0.4$, at time $t=0.001$, for different volume sizes: $R=25$ (empty diamonds), $R=50$ (black triangles), $R=75$ (gray squares), and $R=100$ (dark gray circles). The gray solid lines denote the initial condition.