Effect of boundaries on noninteracting weakly active particles in different geometries

We study analytically how noninteracting weakly active particles, for which passive Brownian diffusion cannot be neglected and activity can be treated perturbatively, distribute and behave near boundaries in various geometries. In particular, we develop a perturbative approach for the model of active particles driven by an exponentially correlated random force (active Ornstein-Uhlenbeck particles). This approach involves a relatively simple expansion of the distribution in powers of the Péclet number and in terms of Hermite polynomials. We use this approach to cleanly formulate boundary conditions, which allows us to study weakly active particles in several geometries: confinement by a single wall or between two walls in 1D, confinement in a circular or wedge-shaped region in 2D, motion near a corrugated boundary, and, finally, absorption onto a sphere. We consider how quantities such as the density, pressure, and flow of the active particles change as we gradually increase the activity away from a purely passive system. These results for the limit of weak activity help us gain insight into how active particles behave in the presence of various types of boundaries.

Figure 1

Several of the geometries we consider: (a) active particles near a single hard wall, (b) active particles inside and outside a circular boundary, (c) active particles confined to a wedge-shaped region, (d) active particles near a corrugated boundary, and, finally, (e) active particles near a spherical absorber.

Figure 2

${\epsilon }^2$ (blue) and ${\epsilon }^4$ (orange) contributions to the density of weakly active particles near a single impenetrable wall [Eq. (18)]. The ${\epsilon }^2$ contribution gives an elevated density while the ${\epsilon }^4$ contribution slightly depletes the density near the wall.

Figure 3

Currents $J_x(x,\eta ),J_{\eta }(x,\eta )$ [Eqs. (20a) and (20b)] resulting from activity near a single impenetrable wall. The circulation shows the simple behavior of active particles propelling toward the wall, spending time turning around, and then propelling away.

Figure 4

Average propulsion $\langle \mathit{\eta }\rangle$ [Eq. (40)] near a right-angled corner that confines active particles to $x>0,y>0$. This orientation toward the corner leads to additional accumulation given by the last term of the density Eq. (38).

Figure 5

${\epsilon }^2$ contributions to the density inside [Eq. (49)] and outside [Eq. (45a)] an impenetrable circular boundary where the density inside the boundary at $r=0$ is maintained to be ${\rho }_{\text{bulk}}$. Here, $R=10\lambda$. Active particles prefer to accumulate on concave surfaces such as the inner side of the circular boundary rather than convex surfaces such as the outer side. This gives rise to a discontinuous drop in density between the inner and outer parts of a circular boundary that is proportional to the curvature $R^{-1}$.

Figure 6

(a) Average propulsion $\langle \mathit{\eta }\rangle$ within a wedge-shaped region with angle $2\alpha =\pi /8$. (b) $x$-component of the average propulsion $\langle {\eta }_x\rangle$ along the center of the wedge $\theta =0$, also for $2\alpha =\pi /8$. This propulsion toward the tip decays rapidly over a length scale $\frac{\lambda }{sin\pi /2^l}$.

Figure 7

(a) Dependence of the correction to wedge tip density ${\mathrm{\Delta }}_{\text{tip}}=\underset{r\to 0}{lim}\mathrm{\Delta }(r,\theta )$ on the wedge angle $2\alpha =\pi /2^{l-1}$ for $l=1,2,\cdots ,15$. The black dotted line shows the linear dependence close to $2\alpha =\pi$, which was obtained using the approach developed in Sec. 6. (b) Weak scaling of ${\mathrm{\Delta }}_{\text{tip}}$ with $2\alpha$ for small wedge angles.

Figure 8

Top: Sawtooth-shaped boundary defined by Eq. (64). Bottom: Currents $\mathit{J}(x,y)=(J_x(x,y),J_y(x,y))$ due to sawtooth-shaped boundaries with asymmetry $\zeta =0$ (left) and $\zeta =0.5$ (right) computed using the first 20 modes of $h\left(x\right)$.

Figure 9

Representation of solid boundaries as ramp potentials.