Noisy pursuit by a self-steering active particle in confinement^(a)

In the limit $\textrm{Pe}\to 0$ and $\Omega \to 0$, the dynamics of a simple AOUP is obtained, which is determined by thermal and active fluctuations, with $\langle {\cos \beta }\rangle =0$, $\langle {v}\rangle =\sqrt {\pi }/2$, and a constant radial distribution function Ψ(r) (see the SM). The presence of steering leads to a competition with active propulsion.

With increasing maneuverability, but $\textrm{Pe}\,\Omega \ll 1$ —steering dominates over noise and active motion—, $|\langle {\cos \beta }\rangle |$ assumes a Pe-independent plateau, which approaches the limiting value $\langle {\cos \beta }\rangle =-1$ for $\Omega \gg 1$, i.e., the propulsion direction is oriented toward the target (fig. S8 in the SM shows trajectories). Our analytical calculations yield the average value (SM, sect. S-III.B)

$$\begin{equation} |\langle{\cos\beta}\rangle| = \frac{I_1(z)}{I_0(z)}, \end{equation} \tag{ 3 }$$

with $z=2\langle {v}\rangle \Omega$, and I₀, I₁ the modified Bessel functions of the first kind. Similarly, for $\Omega |\langle {\cos \beta }\rangle | \gg 1$, the mean velocity approaches a Pe-independent plateau, which is given by

$$\begin{equation} \langle{v}\rangle = \Omega |\langle{\cos\beta}\rangle|, \end{equation} \tag{ 4 }$$

and follows from the general expression (S10) of sect. S-III.A (SM, fig. S1) in the SM. Hence, $\langle {v}\rangle \approx \Omega$ for $|\langle {\cos \beta }\rangle |\lesssim 1$ in agreement with the numerical data. Since $\langle {\cos \beta }\rangle <0$, an increasing maneuverability increases the propulsion velocity according to eq. (2b), and $\langle {v}\rangle$ can exceed v₀ by far. However, this does not imply a more efficient pursuit, since the iAOUP velocity is given by $\dot r$ (eq. (2a)), which is then determined by the translational noise. Taylor expansion of the Bessel functions for large arguments $(\Omega \gg 1)$ and insertion of eq. (4) yields $|\langle {\cos (\beta )}\rangle |\approx 1-1/(4\Omega ^{2})$ for $\Omega >1$, and $|\langle {\cos (\beta )}\rangle |\approx \sqrt {\pi }\,\Omega /2$ for $\Omega \to 0$, in agreement with fig. 2(a). The latter corresponds to the uniform distribution of the bearing angle in absence of steering.

The radial dynamics (eq. (2a)) is predominately diffusive, because thermal noise dominates over the active term. Our analytical calculations yield, with eq. (4), the approximate dependence (sect. S-III.C in the SM)

$$\begin{equation} \langle{r}\rangle = \frac{2}{\langle{v}\rangle \textrm{Pe} \, |\langle\cos\beta\rangle|} = \frac{2 \Omega}{\langle{v}\rangle^2 \textrm{Pe}}. \end{equation} \tag{ 5 }$$

As displayed in fig. S2 in the SM, this expression describes the simulation data very well over a wide range of maneuverability, when the values of $\langle {v}\rangle$ from simulations are inserted. In the limit $\Omega \gg 1$, $\textrm{Pe}\,\Omega <1$, and with $\langle {v}\rangle =\Omega$, we obtain

$$\begin{equation} \langle{r}\rangle = \frac{2}{\textrm{Pe} \, \Omega}, \end{equation} \tag{ 6 }$$

in agreement with simulation results (fig. 2). Thus, the radial distance increases with decreasing Péclet number and maneuverability. Interestingly, the presence of the maneuverability in eq. (6) is in strong contrast to the corresponding behavior of iABPs, where $\langle {r}\rangle$ is independent of Ω for $\textrm{Pe}/\sqrt {\Omega } <1$. The dependence of the propulsion velocity on Ω qualitatively changes the radial distribution of iAOUPs compared to iABPs [8]. Although, the large propulsion velocity, $\langle {v}\rangle \gg 1$, implies an effectively larger Péclet number, the factor $\textrm{Pe}\langle {v}\rangle \langle {\cos \beta }\rangle$ is still much smaller than unity for $\textrm{Pe}\,\Omega \ll 1$, and thermal noise determines the radial distribution function (SM, figs. S6, S8).

For large Pe, two scaling regimes appear, depending on the ratio of $\textrm{Pe}/\Omega$. In the case $\textrm{Pe}\gg \Omega \gg 1$, active propulsion dominates over steering and thermal noise, and the iAOUP behaves as a simple AOUP in the limit $\textrm{Pe}\to \infty$. In particular, the propulsion direction is random (fig. 2(a)), and the bearing angle is determined by active noise, implying a nearly uniform distribution of β and $|\langle {\cos \beta }\rangle |\ll 1$. Similarly, the equation of motion for v is independent of steering for $\Omega |\langle {\cos \beta }\rangle |\ll 1$ and the value $\langle {v}\rangle =\sqrt {\pi }/2$ of a simple AOUP is assumed (fig. 2(b)) (cf. SM, sect. S-III.C). Since $\langle {v}\rangle >1$, and with the assumption $|\langle {\cos (\beta )}\rangle |>0$, the radial distance r is determined by propulsion, and noise can be neglected. Then, we obtain the average radial distance (SM, sect. S-III.C)

$$\begin{equation} \langle{r}\rangle = 2 \frac{\langle{v}\rangle}{\langle{1/v}\rangle} \frac{\textrm{Pe}}{\Omega} = \frac{\textrm{Pe}}{\Omega}, \end{equation} \tag{ 7 }$$

with $\langle {v}\rangle /\langle {1/v}\rangle =1/2$ (SM, eq. (S8)). This power law is in excellent agreement with the simulation data in fig. 2. Not surprisingly, the iAOUP exhibits the same $\textrm{Pe}/\Omega$ dependence as an iABP [8], because the random orientation of the propulsion direction for $\textrm{Pe}\gg \Omega \gg 1$ decouples the propulsion velocity from the other variables, and, on average, the iAOUP behaves similar to an iABP, yet with the mean velocity $\langle {v}\rangle =\sqrt {\pi }/2$ and $\langle {v^{2}}\rangle =1$. However, there is a quantitative difference to an iABP, whose mean radial distance is twice as large. Thus, the fluctuations in the magnitude of the propulsion velocity lead to a closer approach of the iAOUP to the target. The pursuer traverses rosette-like trajectories as displayed in fig. 1(b). Due to the highly persistent motion $(\textrm{Pe}/\Omega \sim v_{0}^{2}/C\gg 1)$, the pursuer overshoots the target, but steering enforces a movement back toward the target.

In the case $\Omega \gg \textrm{Pe}\gg 1$, the dynamics of the propulsion velocity and bearing angle are determined predominately by steering rather than by active propulsion. Our analytical calculations yield the approximated average values (SM, eqs. (S29), (S30))

$$\begin{equation} |\langle{\cos\beta}\rangle| = \frac{1}{\sqrt{\textrm{Pe}\, \Omega}}, \quad\quad \langle{v}\rangle = \sqrt{\frac{\Omega}{\textrm{Pe}}}. \end{equation} \tag{ 8 }$$

These power laws are in very good agreement with the simulation results in figs. 2(a), (b). Interestingly, thermal noise determines the radial dynamics, because $\langle {v}\rangle \textrm{Pe}\langle {\cos \beta }\rangle \approx -1$, and the active term in eq. (2a) is of order unity. Then, the Fokker-Planck equation (S18) in the SM for the radial distance gives $\langle {r}\rangle \approx 2$, in qualitative agreement with simulation results in fig. 2 —the actual value $\langle {r}\rangle$ is slightly smaller, because the approximate exponential distribution deviates somewhat from the proper distribution function (fig. S6 in the SM). The intersection of the power law (6) with $\langle {r}\rangle \approx 2$ suggests that this plateau extends over the range $1/\Omega ^{2} \ll \textrm{Pe}/\Omega \ll 1$. Thus, there is a broad range of Péclet numbers, where a minimum average pursuer-target distance is assumed, independent of Pe and Ω. This is in contrast to iABPs, where the smallest average distance is assumed for $\textrm{Pe}/\sqrt {\Omega }=1$ [8].

The freedom to adopt the propulsion velocity leads to a novel pursuit dynamics, and can provide an advantage of iAOUPs over iABPs, because the minimal average approach is not very sensitive to the actual values of Pe and Ω over a wide range of these values. However, the iAOUP, on average, cannot get arbitrarily close to the target, whereas the iABP minimal average distance $\langle {r}\rangle \sim 1/\sqrt {\Omega }$ decreases with increasing Ω. The intimate coupling of the bearing angle and propulsion velocity allows for an efficient local adaption of propulsion toward the target, where, at a given Pe and an increasing Ω, the bearing angle is increasingly randomized and the average propulsion velocity increases. This leads to a fast exploration of the neighborhood of the target. As a result, the coupling implies an effective thermal-like radial dynamics as in a linearly radial-outward growing potential, i.e., an attractive radial force field (SM, fig. S8).

In the limit $\textrm{Pe}\to 0$, the radial distance is decoupled from the propulsion velocity and steering. The dynamics of r is dominated by confinement and thermal noise, and $\langle {r}\rangle =\sqrt {\pi /(2k)}$ (eq. (9)). Moreover, the propulsion velocity is independent of steering for $\Omega \ll 1$, with $\langle {v}\rangle =\sqrt {\pi }/2$. Both limits are consistent with the simulation data of fig. 3. As in the absence of confinement, the average $\langle {\cos \beta }\rangle$ is given by eq. (3), but with the argument (SM, eqs. (S45), (S46))

$$\begin{equation} z= \frac{\Omega r^2 - v^2 r \textrm{Pe}}{v+ r^2/(2v)}, \end{equation} \tag{ 10 }$$

at constant r and v. For small arguments $z\,(0<\Omega \ll 1,\,\textrm{Pe}=0)$, the Taylor expansion of the Bessel functions yields

$$\begin{equation} \langle{\cos\beta}\rangle \approx - \frac{\Omega \langle{r}\rangle^2}{2 \langle{v}\rangle} = -\Omega \frac{\sqrt{\pi}}{2k}, \end{equation} \tag{ 11 }$$

with the above averages for $\langle {r}\rangle$ and $\langle {v}\rangle$. Thus, we predict $|\langle {\cos \beta }\rangle |$ to decrease linearly with increasing Ω, in agreement with fig. 3(a). With increasing Ω, the tendency of the pursuer to point toward the target increases, and the propulsion velocity becomes Ω dependent. Insertion of eq. (4) into eq. (11) yields $\langle {\cos \beta }\rangle =-\sqrt {\pi /(4k)}$ for $\Omega \gg 1$, a Pe- and Ω-independent plateau, qualitatively consistent with the numerical results (fig. 3(a)). By comparison with simulation results (fig. 4(a)), we find that the k dependence is quantitatively more accurately captured by

$$\begin{equation} \langle{\cos\beta}\rangle = - \frac{1}{\sqrt{2k+1}}, \end{equation} \tag{ 12 }$$

which extends the validity of the average toward small k values. Hence, despite $\Omega \gg 1$, no perfect alignment toward the target is reached, in contrast to an unconfined iAOUP.

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The mean propulsion velocity follows from eqs. (4), (12), and also assumes a Pe-independent plateau, which is given by

$$\begin{equation} \langle{v}\rangle = \frac{\Omega}{\sqrt{2k+1}}, \end{equation} \tag{ 13 }$$

in good agreement with the simulation results. Hence, even the confined pursuer shows an increase of the mean propulsion velocity $\langle {v}\rangle \sim \Omega$ for $\textrm{Pe}\to 0$ (fig. 3(b)), yet the average is reduced by the k-dependent term.

Figure 4 displays the averages $\langle {\cos \beta }\rangle$, $\langle {v}\rangle$, and $\langle {r}\rangle$ as a function of the potential strength k. The approximations in eqs. (9) $(\textrm{Pe}=0)$, (12), and (13) capture the k dependence very well.

In general, $\langle {\cos \beta }\rangle$ increases with increasing k due to confinement, whereas $\langle {v}\rangle$ and $\langle {r}\rangle$ decrease. Stronger confinement reduces the average orientation of the propulsion direction toward the target, and either randomizes it $(\Omega \gg \textrm{Pe})$ or enhances the orientation away from the target $(\textrm{Pe}\gg \Omega )$. This is particularly pronounced for $\textrm{Pe}\gg 1$ and $\Omega <1$, where the propulsion orientation changes from a random value to an orientation away for the target, and for $\textrm{Pe}\ll 1$ and $\Omega \gg 1$, where it changes from a strong orientation toward the target to a more random orientation.

The properties of the pursuer at large Péclet numbers strongly depend on the maneuverability Ω. For $\Omega <1$, the averages $\langle {v}\rangle$, $\langle {\cos \beta }\rangle >0$, and $\langle {r}\rangle$ are close to those of simple AOUPs in the harmonic potential, and decrease with increasing Ω, as displayed in fig. 3. Taylor expansion of the mean propulsion velocity eq. (S10) in the SM for $\Omega \langle {\cos \beta }\rangle \ll 1$ and $\langle {\cos \beta }\rangle >0$ yields (eq. (S51) in the SM)

$$\begin{equation} \langle{v}\rangle = \frac{\sqrt{\pi}}{2}\left(1-\frac{4-\pi}{\sqrt{\pi}}\Omega\langle{\cos\beta}\rangle\right)\!\!. \end{equation} \tag{ 14 }$$

This expression provides the correct limit $\langle {v}\rangle =\sqrt {\pi }/2$ for $\Omega \to 0$ and the propulsion velocity decreases linearly with increasing Ω, in agreement with the numerical results displayed in fig. 3(b).

In the absence of steering, $\Omega =0$, and thermal noise, we extract the dependence (eqs. (S57), (S58) in the SM)

$$\begin{equation} \langle{\cos\beta}\rangle = \langle{\cos\beta}\rangle_0 = 1 - \frac{1}{k+1}, \end{equation} \tag{ 15 }$$

from the Fokker-Planck equation for the radial distance, in agreement with the simulations results even at small k values. Note that $\langle {\cos \beta }\rangle$ is positive, and the propulsion direction points away from the target, which is a consequence of confinement (fig. 3). In the limit $\Omega \ll 1$, we expect $\langle {\cos \beta }\rangle$ to depend linearly on Ω. Hence, to first order in Ω, we can replace $\langle {\cos \beta }\rangle$ in eq. (14) by $\langle {\cos \beta }\rangle _{0}$ (eq. (15)), which gives

$$\begin{equation} \langle{v}\rangle = \frac{\sqrt{\pi}}{2}\left( 1-\frac{4-\pi}{\sqrt{\pi}} \Omega \frac{k}{k+1}\right)\!, \end{equation} \tag{ 16 }$$

again in good agreement with simulations (fig. 3(b)). In the absence of thermal noise, the radial stationary-state Fokker-Planck equation (S47) in the SM yields the mean radial distance (eq. (S56) in the SM)

$$\begin{equation} \langle{r}\rangle = \frac{\langle{v}\rangle\langle{\cos\beta}\rangle }{k} \textrm{Pe}. \end{equation} \tag{ 17 }$$

Figure S4 in the SM shows that $\langle {r}\rangle$ agrees well with the simulation results, when the averages of the simulation data for $\langle {\cos \beta }\rangle$ and $\langle {v}\rangle$ are used. Insertion of the k dependencies of eq. (14) and (15) gives

$$\begin{equation} \langle{r}\rangle = \frac{\sqrt{\pi} \, \textrm{Pe}}{2 k+1} \left( 1 - \frac{4 - \pi}{\sqrt{\pi}} \Omega \frac{k}{k+1}\right)\!, \end{equation} \tag{ 18 }$$

with a correction for small k. The simulation results of fig. 3 confirm the analytical predictions of eq. (16) and (18).

As long as $\Omega \lesssim 1$, steering only weakly affects the properties of the confined pursuer. Steering toward the target enhances the orientation of the propulsion direction toward the target, and consequently, implies a decreasing mean radial distance. The manifestation of the steering correction depends on the strength of the potential, as displayed in fig. 4. Evidently, stronger confinement enhances the radial outward orientation of the propulsion direction. Qualitatively, this behavior is in strong contrast to that in the limit $\textrm{Pe}\to 0$ and $\Omega \gg 1$.

In the limit Pe, $\Omega \gg 1$, and $\textrm{Pe}/\Omega <1$ —here steering dominates over propulsion—, the active term in eq. (2a) exceeds the contribution by the potential. Hence, as in the absence of confinement, $\langle {r}\rangle$ is approximately given by eq. (7), in terms of the averages over the velocity, and (eq. (S62) in the SM)

$$\begin{equation} \langle{\cos\beta}\rangle = - \frac{\langle{1/v}\rangle\Omega}{(\langle{v}\rangle \textrm{Pe})^2} + \frac{2 k}{\langle{1/v}\rangle\Omega}, \end{equation} \tag{ 19 }$$

with an additional contribution by the potential. Applying the approximation $\langle {1/v}\rangle \approx 1/\langle {v}\rangle$ and inserting eq. (4), we obtain

$$\begin{equation} \langle{\cos\beta}\rangle = -\frac{1}{\sqrt{\Omega \textrm{Pe}}} \frac{1}{\sqrt[4]{1+2k}}, \end{equation} \tag{ 20 }$$

and by insertion of eq. (20) into eq. (4)

$$\begin{equation} \langle{v}\rangle = \sqrt{\frac{\Omega}{\textrm{Pe}}} \frac{1}{\sqrt[4]{1+2k}}. \end{equation} \tag{ 21 }$$

As displayed in fig. 3(a), (b), the numerically obtained dependence is well captured by this expression. In addition, eq. (7) yields $\langle {r}\rangle = 2/\sqrt {2k+1}$, a radial distance independent of Pe and Ω. This relation is in qualitative agreement with the mean radial distance (eq. (9)) in the absence of steering and $\textrm{Pe}=0$ for $k \gg 1$. Guided by eq. (9), quantitative agreement with simulations is obtained for the choice (figs. 3(c), 4(c))

$$\begin{equation} \langle{r}\rangle = \sqrt{\frac{\pi}{2k+1}}. \end{equation} \tag{ 22 }$$

Steering dominates propulsion and the pursuer exhibits the same scaling dependence of the considered averages as a free iAOUP. All three expressions exhibit the same Pe and Ω dependence as the unconfined pursuer, yet the magnitude is significantly reduced by the respective k-dependent term (cf. fig. 4).

In the limit $\textrm{Pe}\gg \Omega \gg 1$, propulsion dominates over steering and thermal noise. Nevertheless, steering influences the k-dependent averages (fig. 3). Qualitatively, this follows from the stationary-state Fokker-Planck equations of the respective variables including the potential (SM, sect. S-IV.D.2). We obtain similar approximate scaling relations as in the absence of the potential,

$$\begin{equation} \langle{\cos\beta}\rangle = \ \frac{1}{\Omega} \frac{2 k}{\sqrt{\pi(k+1)}}, \end{equation} \tag{ 23a }$$

$$\begin{equation} \langle{v}\rangle = \frac{\sqrt{\pi}}{2} \frac{1}{\sqrt{k+1}}, \end{equation} \tag{ 23b }$$

$$\begin{equation} \langle{r}\rangle = \frac{\textrm{Pe}}{\Omega (k+1)}, \end{equation} \tag{ 23c }$$

$k\to 0$

$\langle {\cos \beta }\rangle$

$\Omega \langle {\cos \beta }\rangle$

$\langle {v}\rangle$

As displayed in fig. 4, the propulsion direction is random for $k<1$, but points preferentially radially outward with increasing k. In contrast, $\langle {v}\rangle$ and $\langle {r}\rangle$ decrease gradually with increasing k, and $\langle {v}\rangle$ is smaller than $\sqrt {\pi }/2$, the value of a simple AOUP in confinement (fig. 3(b), (c)). However, eq. (23c) applies for $\textrm{Pe}/\Omega \gg (k+1)/\sqrt {k}$ only. Thus, the crossover from the constant plateau $\langle {r}\rangle =\sqrt {\pi /(2k+1)}$ to the universal increase in eq. (23c) shifts to larger $\textrm{Pe}/\Omega \sim \sqrt {k}\,(k\gg 1)$ ratios with increasing k, i.e., the range of the minimal radial distance extends to large $\textrm{Pe}/\Omega$.