Figure 1

Simulation images. Lines indicate barriers and walls and dots indicate particle positions. (a) Initial state of the sample with $r_f=1.0$, $\eta =0.7$, and $r_e=0.07$ at ${\rho }_0=0.4$. (b) The same sample after $7\times {10}^5$ simulation time steps shows rectification of particles into the top chamber.Reuse & Permissions

Figure 2

Density in the top chamber ${\rho }_{\text{top}}$ after $3\times {10}^6$ simulation time steps for a sample with initial density ${\rho }_0=0.4$, indicated by the dashed line. (a) ${\rho }_{\text{top}}$ vs $r_e$ for $\eta =1.5$ and $r_f=1.0$. The rectification reverses at $r_e=0.12$ and drops to zero for $r_e⩾0.3$. (b) ${\rho }_{\text{top}}$ vs $\eta$ for $r_e=0.12$ and $r_f=1.0$. The rectification reverses at $\eta \sim 1.0$ and drops to zero for $\eta \gtrsim \pi$. (c) ${\rho }_{\text{top}}$ vs $r_f$ for $r_e=0.12$ and $\eta =1.5$. For $r_f<r_e$ only steric particle interactions occur and rectification is negligible. For $r_e<r_f<1.2$ rectification is toward the bottom chamber, followed by a rectification reversal at $r_f\approx 1.2$. For large $r_f$, all the particles tend to align into a giant flock and accumulate in the top chamber.Reuse & Permissions

Figure 3

Dependence of rectification behavior on $r_e$, $\eta$, and $r_f$ for a sample with ${\rho }_0=0.4$. (a) Rectification phase diagram for $r_e$ vs $\eta$ with $r_f$ fixed at $r_f=1.0$. Contours indicate lines of constant ${\rho }_{\text{top}}$. Lower contours (red) indicate rectification into the top chamber and upper contours (blue) indicate rectification into the bottom chamber. (b) Rectification curves showing ${\rho }_{\text{top}}$ after $4\times {10}^5$ time steps in the same system as a function of $r_f$ for various selections of $r_e$ and $\eta$: $r_e=0.23$ and $\eta =1.1$ ($⧫$), $r_e=0.11$ and $\eta =2.1$ ($\square$), $r_e=0.09$ and $\eta =1.9$ ($\times$), $r_e=0.07$ and $\eta =1.5$ ($▴$), and $r_e=0.07$ and $\eta =0.7$ ($▽$). The initial density ${\rho }_0=0.4$ is indicated by a dashed line. The location of each curve on the $r_e$ vs $\eta$ phase diagram in (a) is indicated by the corresponding symbol; the point corresponding to the lowest curve ($⧫$) sits at the maximum of reversed rectification and the remaining points ($\square$, $\times$, $▴$, and $▽$) successively ascend the landscape to the peak of forward rectification.Reuse & Permissions

Figure 4

Illustration of rectification into the top chamber for low noise $\eta =1.5$ and small exclusion radius $r_e=0.05$ at $r_f=1.0$ in a sample with ${\rho }_0=0.4$. A $15\times 15$ section of the sample is shown. Symbols denote the following: dots, particle positions; light lines, particle trajectories; and heavy lines, barriers. A flock incident on the gates from the bottom chamber (a) condenses and elongates in order to file through the aperture between gates (b). Flocks incident on the gates from the top chamber have a much lower probability of passing through the aperture and cannot be funneled into a similar oblong shape.Reuse & Permissions

Figure 5

Illustration of rectification into the lower chamber for low noise $\eta =1.5$ and large exclusion radius $r_e=0.2$ at $r_f=1.0$ in a sample with ${\rho }_0=0.4$. A $20\times 20$ section of the sample is shown. Symbols denote the following: dots, particle positions; light lines, particle trajectories; and heavy lines, barriers. (a) and (b) Flocks incident on gates from the bottom chamber cannot fit through the aperture for this value of $r_e$; the flock jams inside the funnel while a single particle (highlighted by the red dot) escapes from the flock and enters the upper chamber. The remainder of the flock returns to the lower chamber. (c)–(e) Flocks incident on gates from the top chamber are fragmented and a small group of particles (highlighted by the red dots) can escape from the flock and enter the lower chamber. The flock fragmentation process occurs with greater frequency as the flocks become less cohesive due to either higher $\eta$ or higher ratios $r_e/r_f$.Reuse & Permissions

Figure 6

(a) Particle density ${\rho }_{\text{top}}$ in the top chamber as a function of time for samples with $\eta =1.5$, $r_f=1.0$, and ${\rho }_0=0.4$. Solid lines denote $r_e=0.05$ and $0.07$, which produce forward rectification, and dashed lines denote $r_e=0.21$ and $0.23$, which produce reverse rectification. (b) Rectification curves for low noise $\eta =1.5$ and exclusion radii $r_e\in [0.01,0.31]$ at $r_f=1.0$ in a sample with ${\rho }_0=0.4$. The density in the top chamber ${\rho }_{\text{top}}$ is shown after $3\times {10}^6$ simulation time steps for $L=33$ (blue $\diamond$), $L=44$ (green $\times$), $L=66$ (yellow $△$), and $L=99$ (brown $◯$). Longer time simulations are also shown: ${\rho }_{\text{top}}$ after $8\times {10}^6$ simulation time steps for $L=33$ (cyan $\square$) and $L=66$ (red $▽$). The inset shows the mean final density in the top chamber under conditions of rectification toward the bottom ${\rho }_{\text{top}-r}={\langle {\rho }_{\text{top}}\rangle }_{0.19<r_e<0.25}$ for various system sizes $L$ and simulation times of $3\times {10}^6$ time steps ($\diamond$) and $8\times {10}^6$ time steps ($*$).Reuse & Permissions

Figure 7

Mean flock size $N_c$, in numbers of particles, vs $r_f$ for $r_e=0.12$ and $\eta =1.5$ (blue circles), vs $r_e$ for $r_f=1.0$ and $\eta =1.5$ (red squares), and vs $\eta$ for $r_f=1.0$ and $r_e=0.12$ (green diamonds). All samples have ${\rho }_0=0.4$. Solid symbols indicate rectification toward the top chamber, while open symbols indicate rectification toward the bottom. Error bars indicate standard deviation.Reuse & Permissions

Figure 8

Dependence of rectification on initial particle number density ${\rho }_0$ for a system with $r_f=1.0$ and $\eta =1.1$. (a) ${\rho }_{\text{top}}/{\rho }_0$ after $3\times {10}^6$ simulation time steps vs $r_e$ for different values of ${\rho }_0$. From blue to red, ${\rho }_0=0.004$ ($\diamond$), $0.01$ ($\square$), $0.03$ ($\times$), $0.05$ ($△$), $0.08$ ($▽$), $0.1$ ($◯$), and $0.12$ (+). A rectification reversal emerges as ${\rho }_0$ increases. (b) ${\rho }_{\text{top}}/{\rho }_0$ after $3\times {10}^6$ simulation time steps vs ${\rho }_0$ for $r_e=0.05$ (top red curve) and $r_e=0.23$ (bottom blue curve).Reuse & Permissions