Velocity Anomaly of a Driven Tracer in a Confined Crowded Environment

We consider a discrete model in which a tracer performs a random walk biased by an external force, in a dense bath of particles performing symmetric random walks constrained by hard-core interactions. We reveal the emergence of a striking velocity anomaly in confined geometries: in quasi-1D systems such as stripes or capillaries, the velocity of the tracer displays a long-lived plateau before ultimately dropping to a lower value. We develop an analytical solution that quantitatively accounts for this intriguing behavior. Our analysis suggests that such a velocity anomaly could be a generic feature of driven dynamics in quasi-1D crowded systems.

Figure 1

A sketch of the confined geometries used in the numerical simulations and in the theoretical modeling (top: stripelike, bottom: capillarylike). Boundary conditions are periodic along directions 2 and 3. The TP, in red, is the only particle subject to the external force $F$.

Figure 2

Main plot: rescaled mean position of the TP as a function of rescaled time ${\rho }_0^{2}t$, from numerical simulations in a 2D stripe with $L=3$ and an infinite external force. The dashed lines are the values of the velocity in the two limit regimes (see main text). Inset: rescaled variance as a function of rescale time for the same system. The solid line is the analytical prediction from [24]. The number of realizations in numerical simulations ranges from ${10}^2$ (for high ${\rho }_0$) to ${10}^5$ (for low ${\rho }_0$).

Figure 3

Rescaled mean position of the TP as a function of rescaled time, for different geometries and forces. (a) 2D stripe with $L=3$ and infinite external force. (b) 2D stripe with $L=3$ and $F=4$, $\beta \sigma =1$. (c) 3D capillary with $L=3$ and infinite external force. For (a),(b), and (c), the solid line is the scaling function $g$ defined in the main text. (d) 2D infinite lattice with infinite drag force. The solid line is the single value $a_0=a_0^{\prime }$. The number of realizations in numerical simulations ranges from ${10}^2$ (for high ${\rho }_0$) to ${10}^5$ (for low ${\rho }_0$).