Driven tracers in narrow channels

Steady-state properties of a driven tracer moving in a narrow two-dimensional (2D) channel of quiescent medium are studied. The tracer drives the system out of equilibrium, perturbs the density and pressure fields, and gives the bath particles a nonzero average velocity, creating a current in the channel. Three models in which the confining effect of the channel is probed are analyzed and compared in this study: the first is the simple symmetric exclusion process (SSEP), for which the stationary density profile and the pressure on the walls in the frame of the tracer are computed. We show that the tracer acts like a dipolar source in an average velocity field. The spatial structure of this 2D strip is then simplified to a one-dimensional (1D) SSEP, in which exchanges of position between the tracer and the bath particles are allowed. Using a combination of mean-field theory and exact solution in the limit where no exchange is allowed gives good predictions of the velocity of the tracer and the density field. Finally, we show that results obtained for the 1D SSEP with exchanges also apply to a gas of overdamped hard disks in a narrow channel. The correspondence between the parameters of the SSEP and of the gas of hard disks is systematic and follows from simple intuitive arguments. Our analytical results are checked numerically.

Figure 1

Scheme of the system studied for large $L_x, L_y=5$, and $Y_T=2$. The small black disks are bath particles that may hop toward neighboring sites with rate 1 if they are empty. The large red disk is the tracer particle that hops with rate $p$ to the right and $q$ to the left. In order to measure the pressure, we introduce an extra site at the boundaries toward which bath particles are allowed to hop with rate $\lambda <1$ (blue dashed-dotted site, also see Sec. 2c2). The extra site always moves with the tracer, so that their distance is constant. In this picture the extra site is in a position to measure the pressure at a distance $(2,3)$ from the tracer.

Figure 2

Density profile along the $x$ direction on each row for a thin channel with $L_y=5$ for fixed $y=0$ (a), $y=1$ (b), and $y=2$ (c). On each graph the results of Monte Carlo (MC) simulations are plotted with red circles, the theoretical expression Eqs. (14, 15, 16, 17, 18, 19, 20, 21, 22, 23) using the approximation Eqs. (25) and (26) with black squares (theory 1), the theoretical expression using the numerically measured values of ${\rho }_{{\mathbf{e}}_{\nu }}, {\mathbf{e}}_{\nu }=\pm {\mathbf{e}}_x,\pm {\mathbf{e}}_y$ Eqs. (14, 15, 16, 17, 18, 19, 20, 21, 22, 23) with blue diamonds (theory 2), and the continuous expression Eq. (12) with a green line. The other parameters are $L_x=81, N=150, p=1.5, q=0.5$, and the tracer is in the middle of the channel, $Y_T=3$. For these parameters the measured values are ${\rho }_{{\mathbf{e}}_x}=0.48, {\rho }_{-{\mathbf{e}}_x}=0.32, {\rho }_{{\mathbf{e}}_y}={\rho }_{-{\mathbf{e}}_y}=0.40$, and $A_{+}=1.77, A_{-}=1.33$, whereas Eqs. (25) and (26) give ${\rho }_{{\mathbf{e}}_x}={\rho }_{-{\mathbf{e}}_x}={\rho }_{{\mathbf{e}}_y}={\rho }_{-{\mathbf{e}}_y}=\frac{150}{81\times 5-1}\simeq 0.37, A_{+}=1.94$, and $A_{-}=1.31$.

Figure 3

Tracer velocity ${\mathcal{V}}_{\mathrm{tr}}^{2\mathrm{DSSEP}}$ as a function of the bias $p-q$ for different system sizes. Monte Carlo measurements (pluses, the curves move up as the width $L_y$ increases) are compared to theory (3), Eqs. (14, 15, 16, 17, 18, 19, 20, 21, 22, 23) (solid lines). The length of the channel $L_x=41$ and the global density is held approximately constant, $\overline{\rho }\simeq 0.81$.

Figure 4

Measured ${\rho }_W\left(\lambda \right)$ for a system of size $L_x=81, L_y=9$, with the tracer in the channel center, with $Y_T=5$. The pressure is obtained at a distance $(x,4)$ from the tracer, where the curves are shown for $x=-5, -3, -1$, and 1, from bottom to top. The hopping parameters of the tracer are $p=1.9$ and $q=0.1$, and there are $N=365$ particles in the system.

Figure 5

Monte-Carlo (red circles) and theoretical (black squares) values of the pressure for the same system as in Fig. 4. The value of the background pressure is $-log(1-\overline{\rho })\simeq -log\left(1-\frac{365}{9\times 81-1}\right)\simeq 0.696$.

Figure 6

Left panel: Two possible configurations of the system and some allowed transitions. The black disks are the bath SSEP particles, and the large red disk is the tracer. Bath particles hop symmetrically toward the right and left with rate 1 on each side if their target site is empty. The tracer hops to the right with rate $p$ and to the left with rate $q$ if its target site is empty, as is shown by the top scheme. A tracer and a neighboring particle exchange their positions with rate $\epsilon p$ or $\epsilon q$, if the tracer moves to the right or to the left, respectively. This may occur in the configuration at the bottom of the left panel. Right panel: Configuration of the zero-range process (ZRP) equivalent to the top left SSEP for the case $\epsilon =0$. The tracer is mapped to a special link (red tick), where the transfer rates are $q$ when the ZRP particle hops to the right, and $p$ when the particle hops to the left as indicated by the purple and blue arrows.

Figure 7

Bath particle and total currents ${\mathcal{J}}_{\mathrm{B}}$ (blue line and symbols, bottom) and $\mathcal{J}$ (red line and symbols, top) as a function of $\epsilon$ for $p=1.3, q=0.7, N=99$, and $L_x=500$. The theoretical curves are given by Eqs. (59) and (60).

Figure 8

Velocity of the tracer, ${\mathcal{V}}_{\mathrm{tr}}^{1\mathrm{DSSEP}}$, as a function of $\epsilon$ for $p=1.3, q=0.7, N=99$, and $L_x=500$. The theoretical curve is given by Eqs. (58). Inset: Same for small values of $\epsilon$. Equation (58) (in black solid line) is compared to Eq. (61) (blue dashed line).

Figure 9

Local pressure $P_{yy}$ (dots) and local density $\rho$ (smooth lines) as a function of the separation $x$ from the tracer for various external forces $F=0.1,0.2$, and 0.4. The channel is of length $L_x=1000$ and of width $L_y^{\prime }=2.05$. 99 neutral particles and a single tracer at a temperature $k_{B}T=1$ are considered.

Figure 10

Coarse-grained motion of the tracer along $X$. The continuous curve is the actual position of the center of the tracer, while the step-like line is the coarse-grained picture of a tracer hopping from site to site. For the counting of the up- and down-steps (see the main text) the periodicity of the boundary in $X$ direction needs to be unfolded. No neutral particle is required.

Figure 11

Local density in the comoving frame of the tracer for various $N=49,79,99$ (from bottom to top). Solid lines represent the HD system and dotted lines are the SSEP results. The channel widths $L_y^{\prime }=1.90$. Particles cannot pass each other, and $\epsilon =0$. The driving field $F=0.2$. The length of the periodic channel $L_x=1000$. The associated data is found in Table 1.

Figure 12

Evolution of a tracer particle and of a neutral particle along the channel in $X$ direction as a function of time $t$. The channel width $L_y^{\prime }=2.1$. Thus, overtake events occur at times indicated by the vertical arrows. Here, the length of the periodic channel ($L_x$ is 50) ranges from $X=-25$ to $X=25$.

Figure 13

Local density in the comoving frame of the tracer for the same HD systems as in Table 2 (solid lines), compared to their SSEP equivalents given in the table (dotted lines). The widths are $L_y^{\prime }=2.02$, 2.05, and 2.10 from bottom to top in the $x<0$ region.