Dynamics and density distribution of strongly confined noninteracting nonaligning self-propelled particles in a nonconvex boundary

We study the dynamics of nonaligning, noninteracting self-propelled particles confined to a box in two dimensions. In the strong confinement limit, when the persistence length of the active particles is much larger than the size of the box, particles stay on the boundary and align with the local boundary normal. It is then possible to derive the steady-state density on the boundary for arbitrary box shapes. In nonconvex boxes, the nonuniqueness of the boundary normal results in hysteretic dynamics and the density is nonlocal, i.e., it depends on the global geometry of the box. These findings establish a general connection between the geometry of a confining box and the behavior of an ideal active gas it confines, thus providing a powerful tool to understand and design such confinements.

Figure 1

Notations for a particle at the wall. The particle is characterized by its arclength $s$ along the wall and its orientation $\widehat{\mathit{\nu }}=cos\theta \widehat{\mathbf{x}}+sin\theta \widehat{\mathbf{y}}$. The local normal to the wall is $\widehat{\mathbf{n}}=cos\psi \widehat{\mathbf{x}}+sin\psi \widehat{\mathbf{y}}$.

Figure 2

Dynamics in a confining box in the absence of angular noise. Top left: real-space representation. The concave region (between $B$ and $C$) is shown in orange (gray). The arclength $s$ is counted counterclockwise. The points $A$ and $D$ have the same normal (shown as a straight arrow) as $C$ and $B$, respectively. Top right: normal angle $\psi$ as a function of arclength $s$. The dynamics of a particle with constant orientation $\theta$ is controlled by the distance $\varphi =\theta -\psi$ between the curve and the horizontal dashed line. The fixed points are shown as filled (stable) and empty (unstable) squares. Bottom left: gliding velocity $\dot{s}=v_{0}sin\varphi$ for a particle with orientation $\widehat{\mathit{\nu }}$ (shown inside the box). The orientation and fixed points are the same as those shown in the top right panel. Arrow heads indicate the gliding direction. The colorbar is shown on the right.

Figure 3

Three representations of a nonconvex box showing the concave regions in orange (gray) and the jumps over the concave regions (curved arrows). Convex regions are indexed by numbers 1–3. Region $i$ is delimited by the two inflection points $A_{2i-1}$ and $A_{2i}$. Each $A_i$ is the starting point of a jump over the neighboring concave region that lands at $B_i$, which has the same normal angle ${\psi }_i$ as $A_i$. Top: shape of the box. The “$\times$” on the right is the arclength origin, $s=0$. Bottom left: normal angle vs arclength. Bottom right: normal angle representation. The vertical axis labels the convex region corresponding to each interval. Region 1 appears split into two parts due to periodic boundary conditions.

Figure 4

Density of particles in normal angle space as a function of the normal angle $\psi$ in each of the three convex lobes of the box shown in Fig. 3. ${\psi }_i$ is the normal angle of the inflection point $A_i$ as well as that of its corresponding landing point $B_i$ (see Fig. 3).

Figure 5

Density of particles per unit length of boundary as a function of arclength $s$ for the box shown in Fig. 3. $s_i$ and $s_i^{\prime }$ are the arclengths of the inflection points $A_i$ and corresponding landing points $B_i$ (see Fig. 3). Right panel: heat map for the density along the boundary, shown in real space.

Figure 6

Three representations of a nonconvex box showing the concave regions in orange (gray) and the jumps over the concave regions (curved arrows). Convex regions are indexed by numbers 1–4. Top: shape of the box. The origin of arclengths is the rightmost point on the boundary, halfway between $B_3$ and $B_8$. Bottom left: normal angle vs arclength. Bottom right: normal angle representation. The vertical axis labels the convex region corresponding to each interval. Regions 1, 2, and 4 appear split into two parts due to periodic boundary conditions.

Figure 7

Density of particles in normal angle space as a function of the normal angle $\psi$ in each of the four convex lobes of the box shown in Fig. 6.

Figure 8

Density of particles per unit length of boundary as a function of arclength $s$ for the box shown in Fig. 6. Right panel: Heat map for the density along the boundary, shown in real space.

Figure 9

Positions of peaks in the distribution of the orientations $P\left(\varphi \right)$ relative to the normal as a function of the arclength $s$ over one lobe of the box defined by Eq. (14) with $k=4$ and $r_1=0.5$. The lobe geometry is shown in the right panel. The angular diffusion constant is $D_{\mathrm{r}}={10}^{-4}$, deep in the strong confinement regime. The simulation data are shown as circles. The theoretical prediction is shown as lines. Colors denote the origin of the particles: blue for particles jumping from $A_8$ (upper gray curve), orange for particles jumping from $A_3$ (lower gray curve), and black for particles from the lobe under consideration. Dotted lines represent the paths of particles that fly through the bulk.

Figure 10

The density observed in simulations at various positions along the boundary is plotted against the predicted density for six boxes from the family defined by Eq. (14) and for four values of the angular diffusion constant $D_{\mathrm{r}}$. The dashed line corresponds to a density equal to the predicted density. The symbol associated with each box geometry is shown at the center of that box on the right side.

Figure 11

Density in normal angle space in the convex region shown in Fig. 9 for several values of the angular diffusion constant $D_{\mathrm{r}}$ from deep in the strong confinement regime ($D_{\mathrm{r}}={10}^{-4}$) to outside of it ($D_{\mathrm{r}}={10}^{-1}$). The dashed line is the theoretical prediction from Sec. 4c2 [see Eq. (12)], corresponding to $D_{\mathrm{r}}\to 0$.

Figure 12

Types of jumps over a concave region. The right column shows the orientation $\varphi ={\psi }_B-\psi$ relative to the boundary as a function of the arclength $s$ (counted counterclockwise). Concave regions are in orange (gray). Trajectories through the interior of the box are shown as arrows. $B$ and $C$ are the inflection points delimiting the concave region. $D$ is the first location after $C$ with the same normal as $B$. Top row: the particle follows the boundary and stops at $D$. See Appendix pp2 for the definition of regions (i), (ii), and (iii). The dashed (in green) and the dot-dashed (in blue) curves correspond to a quadratic expansion near $B$ and a linear expansion near $D$, respectively. Middle row: when ${\psi }_B-{\psi }_C>\pi /2$, the particle leaves the boundary at $E$ where ${\psi }_E={\psi }_B-\pi /2$. It travels in a straight line through the bulk until $F$, then follows the boundary to $D$. Bottom row: when ${\psi }_B-{\psi }_C>\pi /2$ and there are multiple concave regions, the particle may fly past $D$ to end its jump in a non-neighboring convex region at a point $D^{\prime }$ with the same normal as $B$.