Swimmer dynamics in externally-driven fluid flows: The role of noise

The advection of self-propelled particles in externally-driven fluid flows presents many surprises when compared to passive advection. Perhaps the biggest surprise is that the transport efficiency of swimmers does not simply increase as swimming speed increases. For example, when swimmers are placed in a two-dimensional (2D) oscillating vortex array exhibiting chaotic mixing, faster swimming does not always lead to faster transport Khurana2011 ; Khurana2012 . Even in a steady 2D vortex array, swimmer trapping inside vortices may be enhanced when the particles swim faster, depending on the shape of the particle Berman2020 . Similarly, transport efficiency does not simply increase as a swimmer’s rotational diffusivity increases, either. In fact, the opposite occurs in the 2D oscillating vortex array Torney2007 ; Khurana2012 . It is reasonable to expect that the more a swimmer’s propulsion direction fluctuates, the smaller its net displacement in a fixed amount of time, and hence the lower the transport efficiency. Unexpectedly however, the presence of both rotational noise and shear flow can effectively trap swimmers in certain regions, as has been experimentally observed for swimming bacteria Rusconi2014 and swimming phytoplankton Barry2015 in a channel flow. While numerous studies have investigated the spatial distributions of noisy swimmers in a variety of flows Santamaria2014 ; Bearon2015 ; Ezhilan2015 ; Vennamneni2020 ; Dehkharghani2019 , a basic understanding of how rotational noise alters swimmer trajectories is lacking. Our objective in this paper is to develop a theory that quantifies the effect of noise on swimmer dynamics in externally driven fluid flows, especially near transport barriers.

Recently, transport barriers analogous to separatrices—and the related invariant manifolds—of passive advection were identified for self-propelled particles in fluid flows Berman2021a . Perfectly smooth-swimming particles are blocked by so-called swimming invariant manifolds (SwIMs) in position-orientation space. The SwIMs project to one-way barriers, called SwIM edges, to swimmer motion in position space. Swimmers with orientational noise, on the other hand, can cross SwIM edges, but they are still blocked by one-way barriers known as burning invariant manifolds (BIMs), which were originally introduced as barriers for propagating chemical reaction fronts in fluid flows Mahoney2012 ; Mitchell2012 . By one-way barriers, we mean that swimmers can pass through a BIM or SwIM edge in one direction, but not the other.

This theory was applied to analyze the experimental trajectories of smooth-swimming and run-and-tumble Bacillus subtilis bacteria in a microfluidic cross-channel featuring a hyperbolic fluid flow, illustrated in Fig. 1a. Whereas the run-and-tumble bacteria exhibit strong rotational noise in the form of sporadic, abrupt changes in swimming direction (Fig. 1c), the smooth-swimming bacteria tend to swim straight in the absence of a flow, with minimal rotational noise (Fig. 1b). The vertical lines in Figs. 1b and 1c are the SwIM edges blocking inward swimming particles, while the horizontal lines are the SwIM edges blocking outward swimming particles. In the hyperbolic flow, the BIMs coincide exactly with the SwIM edges, and hence these curves are one-way barriers for both the smooth swimming and run-and-tumble bacteria. Here, all experimentally recorded trajectories are rectified so they appear to enter the flow from above and exit to the right. Therefore, there can be no trajectories to the left of the SwIM edge $x=-1$, beyond which any trajectory would be swept to the left, as is evident from Figs. 1b and 1c. However, we observe a gap between the left SwIM edge and the measured trajectories of run-and-tumble bacteria in Fig. 1c, compared to the smooth-swimmers in Fig. 1b that can just graze the left SwIM edge before swimming off to the right. Because the gap represents a depletion of the density of trajectories near the SwIM edge relative to the smooth swimmer case, we refer to it as the depletion effect. The depletion effect is caused by the orientation fluctuations of the run-and-tumble bacteria. A run-and-tumble swimmer near this SwIM edge and initially swimming to the right is very likely to tumble and end up crossing the left SwIM edge, forcing it to escape to the left. At the same time, we observe that the run-and-tumble bacteria swim much closer to the lower SwIM edge than their smooth-swimming counterparts. This again is due to tumbling. While smooth swimmers get aligned with the extensional $x$ direction of the flow and thus cannot swim very far below the line $y=0$, run-and-tumble swimmers can tumble out of alignment and swim towards the lower SwIM edge. These stark differences between the trajectories of smooth versus run-and-tumble swimmers have motivated the present work.

In this paper, we show how to calculate the probability of particular noisy swimmer trajectories in a given fluid flow, taking the hyperbolic flow as a case study. Our approach focuses on computing solutions to the time-dependent Fokker-Planck equation of a swimmer (alternately known as a master equation or Smoluchowski equation) in the weak-noise limit Graham1984 ; Dykman1996 ; Gaspard2002 ; Nolting2016 ; Gu2020 . This differs from traditional approaches to the swimmer Fokker-Planck equation, which are focused on the stationary (time-independent) solution and are in the Eulerian frame-of-reference Rusconi2014 ; Bearon2015 ; Ezhilan2015 ; Vennamneni2020 . In contrast, we construct a time-dependent swimmer probability density function by following the Lagrangian paths of a swimmer. This procedure is derived from the weak-noise limit in a manner that is nearly identical to the semiclassical approximation in quantum mechanics Littlejohn1992 , so we refer to it as the semiclassical approximation to the Fokker-Planck equation. We use this approach to quantify the depletion of noisy swimmers near a BIM, and compare the results of our semiclassical calculations with Monte Carlo calculations, i.e. direct numerical simulations of the swimmer equations of motion.

The paper is organized as follows. In Sec. II, we provide background information on the model for swimmer motion employed here and the semiclassical approximation to the Fokker-Planck equation. In Sec. III, we review the dynamics of a deterministic smooth swimmer in the hyperbolic flow, in particular the role of the SwIMs and BIMs. In Sec. IV, we compute the position-independent orientation distributions of a swimmer in the hyperbolic flow, obtaining results analogous to the orientation distributions of magnetotactic and viscotactic swimmers in external fields Waisbord2016 ; Rupprecht2016 ; Stehnach2021 , and we apply the semiclassical approximation to calculate the orientation distribution of swimmers with both rotational diffusion and run-and-tumble dynamics. In Sec. V, we compare Monte Carlo and semiclassical calculations of the depletion effect. Concluding remarks are in Sec. VI. In the appendix, we present a complete derivation of the semiclassical approximation to the Fokker-Planck equation.

We consider the motion of an ellipsoidal swimmer in two dimensions, with position ${\bf r}=(x,y)$ and orientation $\hat{{\bf n}}=(\cos\theta,\sin\theta)$. The stochastic differential equations describing a noisy swimmer in a fluid flow ${{\bf u}}({\bf r})$ are Torney2007 ; Khurana2012

		$\displaystyle{\rm d}{\bf r}=\left[{\bf u}({\bf r})+v_{0}\hat{{\bf n}}\right]{\rm d}t+\sqrt{2D_{T}}\,{\rm d}w_{\bf r},$		(1a)
		$\displaystyle{\rm d}\theta=\left[\frac{\omega({\bf r})}{2}+\alpha\hat{{\bf n}}_{\perp}\cdot{\mathsf{E}}({\bf r})\hat{{\bf n}}\right]{\rm d}t+\sqrt{2D_{R}}\,{\rm d}w_{\theta}+{\rm d}L(\nu),$		(1b)

where $\omega=\hat{\bf z}\cdot(\nabla\times{\bf u})$ is the vorticity, $\hat{{\bf n}}_{\perp}=(-\sin\theta,\cos\theta)$, and ${\mathsf{E}}=(\nabla{\bf u}+\nabla{\bf u}^{\rm T})/2$ is the symmetric rate-of-strain tensor. The shape parameter $\alpha$ equals $(a^{2}-1)/(a^{2}+1)$, where $a$ is the aspect ratio of the ellipse; $\alpha$ varies from $-1$ to $1$, where $\alpha=0$ is a circle, and $|\alpha|=1$ is an infinitely thin rod. Positive (negative) values of $\alpha$ correspond to swimming parallel (perpendicular) to the major axis. Each equation of (1) contains a deterministic drift term, proportional to ${\rm d}t$, and noise terms. In Eq. (1a), the noise terms are the independent Wiener processes $w_{\bf r}=(w_{x},w_{y})$ and account for translational diffusion with diffusivity $D_{T}$. Note that for certain swimmers, the strength of the translational diffusivity along the particle’s major axis may differ from the translational diffusivity along the minor axis, and their may be additional correlations between translational and rotational noise Thiffeault2021 . We ignore these issues here for simplicity.

Equation (1b), on the other hand, contains two stochastic terms describing fluctuations in the swimming direction. We distinguish between two types of rotational noise: rotational diffusion and tumbling. Rotational diffusion refers to continual random perturbations in the swimmer orientation, such that in the absence of a flow, the orientation $\theta$ would exhibit free diffusion (given by the Wiener process $w_{\theta}$) with a rotational diffusivity $D_{R}$. This can arise due to random fluctuations in the propulsion force of the swimmer Hyon2012 ; Thiffeault2021 or from the thermal fluctuations of the surrounding fluid. In the former case, the noise mechanism would lead to correlations between translational and rotational diffusion, but here we neglect those for simplicity. Tumbling refers to the sudden resetting of $\theta$ to a random orientation, independent of its present value, which occurs sporadically as a Poisson process $L(\nu)$ with frequency $\nu$. This kind of sudden, random reorientation is seen in swimming bacteria in the “run-and-tumble” mode of swimming. In practice, the distribution of new orientations may depend on the previous value, as is the case for wild-type strains of the swimming bacteria E. coli Berg1972 , but we neglect this here for simplicity. Hence, the new angle after a tumble is uniformly and randomly distributed between $0$ and $2\pi$.

Our goal in this paper is to estimate the probability of various swimmer trajectories of Eq. (1). This can certainly be accomplished by Monte Carlo simulations, i.e. direct numerical simulations of Eq. (1), but we also develop an analytical approach to calculating such probabilities, which is less computationally costly and provides deeper theoretical insight into the swimmer dynamics. To study the probability of swimmer trajectories, we investigate the Fokker-Planck equation for the probability density of the particle $P({\bf r},\theta,t)$ Solon2015 ,

$$\frac{\partial P}{\partial t}=-\nabla\cdot({\bf F}P)+\frac{\varepsilon}{2}\left[\gamma\left(\frac{\partial^{2}P}{\partial x^{2}}+\frac{\partial^{2}P}{\partial y^{2}}\right)+\frac{\partial^{2}P}{\partial\theta^{2}}\right]+\lambda\left[-P+\frac{1}{2\pi}\int_{0}^{2\pi}P({\bf r},\theta^{\prime},t){\rm d}\theta^{\prime}\right],$$

(2)

where we have abbreviated the deterministic drift terms from Eq. (1) as ${\bf F}$, with

$${\bf F}=\left(u_{x}+v_{0}\cos\theta,\,\,u_{y}+v_{0}\sin\theta,\,\,\frac{\omega}{2}+\alpha\hat{\bf n}_{\perp}\cdot{\mathsf{E}}\hat{\bf n}\right).$$

(3)

Here, $\nabla=(\partial/\partial x,\partial/\partial y,\partial/\partial\theta)$. We have non-dimensionalized the coordinates using a length scale $\mathcal{L}$ and velocity scale $\mathcal{U}$, so that

$$\varepsilon=\frac{2D_{R}\mathcal{L}}{\mathcal{U}}$$

(4)

is the strength of rotational diffusion, $\gamma=D_{T}/(\mathcal{L}^{2}D_{R})$ is the ratio of translational diffusion to rotational diffusion (usually $\gamma<1$), and $\lambda=\nu\mathcal{L}/\mathcal{U}$ is the non-dimensional tumbling rate. Note that the rotational Péclet number is ${\rm Pe}=2/\varepsilon$ Ezhilan2015 . The first two terms on the right-hand side of Eq. (2) are the usual drift and diffusion terms. The third term proportional to $\lambda$ accounts for tumbling, with the first term in brackets describing the loss of probability due to tumbling out of the present angle, and the second term describing the gain of probability from the swimmers at all other angles that have tumbled into the present angle. Equation (2) is difficult to attack in general, so we focus on special cases where exact or approximate analytical (or semi-analytical) solutions may be found.

Of particular interest is the $\lambda=0$ case, describing non-tumbling swimmers or, alternatively, the evolution of the probability density in between tumble events. In this case we seek asymptotic solutions in the weak diffusion ($\varepsilon\ll 1$) limit, which have the WKB form

$$P({\bf r},\theta,t)\approx A({\bf r},\theta,t)\exp\left[-\frac{W({\bf r},\theta,t)}{\varepsilon}\right].$$

(5)

Equation (5) has the same form as the semiclassical approximation to the wave function in quantum mechanics, and hence we refer to it as the semiclassical approximation. Substituting this approximation into Eq. (2) with $\lambda=0$ leads to a Hamilton-Jacobi equation for $W$ and a related transport equation for $A$.

Here, we briefly describe the mathematical theory of approximation (5), while a detailed derivation and discussion are contained in Appendix A. The solution $W$ to the Hamilton-Jacobi equation (52) is related to the classical action accumulated along the trajectories of a particular Hamiltonian system associated with Eq. (2). The Hamiltonian is given by

$$H({\bf q},{\bf p})=\frac{1}{2}\left[\gamma(p_{x}^{2}+p_{y}^{2})+p_{\theta}^{2}\right]+{\bf p}\cdot{\bf F}({\bf q}),$$

(6)

where ${\bf q}=(x,y,\theta)$ and ${\bf p}=(p_{x},p_{y},p_{\theta})$. Equation (6) follows from the Hamiltonian (53) for a general Fokker-Planck equation (43), of which Eq. (2) (with $\lambda=0$) is a special case. The function $W$ is equivalent to the Onsager-Machlup-Freidlin-Wentzell action function Onsager1953a ; Freidlin2012 ; Gaspard2002 which arises in nonequilibrium statistical mechanics Graham1984 ; Dykman1996 and rare event modeling Forgoston2018 . At each point $({\bf r},\theta)$, the action $W({\bf r},\theta,t)$ can be expressed as an integral along the trajectory of the Hamiltonian system with Hamiltonian (6) that arrives at that point at time $t$. This makes Eq. (5) a Lagrangian, as opposed to Eulerian, description of the probability density. The trajectories associated with the minima of $W$, i.e. the minimum-action paths, correspond to the most likely trajectories of a noisy swimmer, because at lowest order in $\varepsilon$, the probability density (5) is peaked at these points. Hence, finding the minimum-action paths is the main focus of most works involving the Onsager-Machlup-Freidlin-Wentzell action function, including recent work on escape paths of active particles in potential wells Gu2020 . In contrast, we consider all possible paths, in order to get a global approximation to the probability density (5).

Throughout the paper, we focus on the hyperbolic flow ${\bf u}=(Bx,-By)$. Therefore, Eq. (1) becomes

		$\displaystyle{\rm d}x=(x+\cos\theta){\rm d}t+\sqrt{\varepsilon\gamma}{\rm d}w_{x},$		(7a)
		$\displaystyle{\rm d}y=(-y+\sin\theta){\rm d}t+\sqrt{\varepsilon\gamma}{\rm d}w_{y}$		(7b)
		$\displaystyle{\rm d}\theta=-\alpha\sin(2\theta){\rm d}t+\sqrt{\varepsilon}{\rm d}w_{\theta}+{\rm d}L(\lambda).$		(7c)

We have taken the velocity scale $\mathcal{U}=v_{0}$ and the length scale $\mathcal{L}=v_{0}/B$ in the non-dimensional equation (7). The typical values of $\varepsilon$, $\gamma$, $\lambda$, and $\alpha$ depend on the system being considered. In the hyperbolic flow experiments leading to Fig. 1, $B=0.44\,\,{\rm s}^{-1}$ Berman2021a . The measured rotational diffusivity for several species of swimming phytoplankton is $D_{R}=0.15$–$0.27\,\,{\rm rad}^{2}/{\rm s}$ Barry2015 , which in the hyperbolic flow would yield $\varepsilon=0.68$–$1.2$. For wild-type E. coli, which exhibit run-and-tumble behavior, the rotational diffusivity during runs is $D_{R}=0.06\,\,{\rm rad}^{2}/{\rm s}$ Locsei2009 and the tumbling frequency is approximately $\nu=1\,\,{\rm s}^{-1}$ Berg1972 , which in the hyperbolic flow would yield $\varepsilon=0.27$ and $\lambda=2.3$. Assuming that the translational diffusion for wild-type E. coli is due only to thermal fluctuations so that $D_{T}=0.2\,\,\mu{\rm m}^{2}/{\rm s}$ Bearon2015 and a swimming speed $v_{0}=14\,\,\mu{\rm m}/s$ Berg1972 , the translational-to-rotational diffusion ratio would be $\gamma=0.003$. Note that in the hyperbolic flow, $\varepsilon$ and $\lambda$ can always be made smaller and $\gamma$ larger by increasing $B$, the flow strength parameter. We take $\alpha=1$ in all numerical computations, corresponding to the elongated shape of swimming bacteria like E. coli and B. subtilis. Numerical solutions of Eq. (7) are obtained using the Euler-Maruyama method. Before investigating the dynamics of noisy swimmers in the hyperbolic flow, we study the deterministic dynamics of Eq. (7) with $\varepsilon=0$ and $\lambda=0$.

The deterministic dynamics of Eq. (7) is best understood through the system’s fixed points and invariant manifolds, previously studied in Ref. Berman2021a . The system possesses four fixed points, which we refer to as swimming fixed points (SFPs) to distinguish them from the passive fixed points of the fluid flow. Denoting the phase-space coordinate ${\bf q}=(x,y,\theta)$, the fixed points are

$${\bf q}_{\pm}^{\rm out}=\left(0,\pm 1,\pm\frac{\pi}{2}\right),\quad{\bf q}^{\rm in}_{+}=(1,0,\pi),\quad{\bf q}^{\rm in}_{-}=(-1,0,0),$$

(8)

illustrated in Fig. 2. Each of the SFPs is a saddle. When $\alpha>0$, and in particular when $\alpha=1$, the ${\bf q}_{\pm}^{\rm out}$ fixed points have stable-unstable-unstable (SUU) linear stability, and the ${\bf q}_{\pm}^{\rm in}$ fixed points have stable-stable-unstable (SSU) linear stability. Hence, the ${\bf q}_{\pm}^{\rm out}$ SFPs possess 2D unstable manifolds, while the ${\bf q}_{\pm}^{\rm in}$ fixed points possess 2D stable manifolds. We refer to these 2D manifolds as swimming invariant manifolds (SwIMs), to distinguish them from the invariant manifolds of passive advection Berman2021a .

Taken as a whole, the stable and unstable SwIMs consist of two interlocking S-shaped sheets, plotted in Fig. 2a. The stable SwIMs (the blue surface) attached to ${\bf q}_{+}^{\rm in}$ and ${\bf q}_{-}^{\rm in}$ share common boundaries along the lines $\{(x,y,\theta)\,\,|\,\,x=0,\,\,\theta=\pm\pi\}$ which are the 1D stable manifolds of ${\bf q}_{\pm}^{\rm out}$ (dark blue lines). Hence, the union of the 2D stable SwIMs with the 1D stable manifolds of ${\bf q}_{\pm}^{\rm out}$ is a surface (blue surface in Fig. 2a) which separates phase space into two pieces. By symmetry, the same geometric shape can be constructed by taking the union of the unstable SwIMs of ${\bf q}_{\pm}^{\rm out}$ with the 1D unstable manifolds of ${\bf q}_{\pm}^{\rm in}$, leading to the red surface in Fig. 2a. The shape of the stable SwIMs is independent of the $y$ coordinate and, similarly, the shape of the unstable SwIMs is independent of the $x$ coordinate. This occurs because in the hyperbolic flow, the $x\theta$ equations Eq. (7a) and (7c) are decoupled from $y$ and similarly the $y\theta$ equations Eq. (7b) and (7c) are decoupled from $x$. The stable and unstable SwIMs intersect along heteroclinic orbits going from one fixed point to another, indicated by the yellow curves in Fig. 2a.

Cross-sections of the SwIMs are shown in Fig. 3, along with the phase portraits of the $x\theta$ dynamics and the $y\theta$ dynamics. Figure 3a shows that swimmers on the left of the stable SwIM ultimately exit the flow to the left, while swimmers on the right ultimately exit right. Similarly, the unstable SwIM (Fig. 3b) separates swimmers that entered the flow from the top from those which entered from the bottom. The SwIMs are therefore transport barriers to swimmers in the hyperbolic flow, because they carve out the $xy\theta$ phase space into distinct, qualitatively different families of trajectories. Importantly, these barriers are nonporous in phase space, meaning no swimmer trajectory can cross them (in the absence of noise).

On the other hand, in position space, the SwIMs project to one-way barriers, allowing swimmers to cross in one direction but not the other. Figure 2b shows the singularities of the projection of the SwIMs into the $xy$ plane—that is, the folds of the S-shapes which bound the projection of the 2D surfaces into the plane. We refer to these curves as SwIM edges Berman2021a . The stable SwIM edges at $x=\pm 1$ (blue curves in Fig. 2b) block inward swimming particles, while allowing outward swimming particles through. To see this, note that for $x=-1$, $\dot{x}\leq 0$ for all $\theta$, and for $x=1$, $\dot{x}\geq 0$ for all $\theta$, as shown in Fig. 3a. Along the stable SwIM edges, the outward fluid flow overpowers the swimmers and they are swept away from the center of the flow. Similarly, the unstable SwIM edges (red curves in Fig. 2b) block outward swimming particles, while inward swimming particles can pass through them. Here, for $y=1$, $\dot{y}\leq 0$, and for $y=-1$, $\dot{y}\geq 0$. On the unstable SwIM edges, it is the inward flow which overpowers the swimmers and pushes them towards the center of the flow.

The SwIM edges in the hyperbolic flow coincide exactly with the BIMs—the 1D invariant manifolds of the SFPs when $\alpha=-1$ Berman2021a ; Mahoney2015 . This is important because SwIM edges are only guaranteed to be one-way barriers for purely deterministic swimmers. BIMs, on the other hand, have stronger barrier properties, in that they are also one-way barriers for swimmers with rotational diffusion or run-and-tumble dynamics in the limit of negligible translational diffusion Berman2021a . Thus, in the hyperbolic flow, the SwIM edges also act as one-way barriers for swimmers with rotational noise. This explains why the run-and-tumble bacteria in the hyperbolic flow experiment remain bounded by the unstable SwIM edge at $y=-1$ in Fig. 1b. Similarly, the stable SwIM edges act as points of no return for all swimmers. Once a swimmer swims over a stable SwIM edge, it is unable to swim back to the center of the flow. This is the origin of the depletion effect we observe when comparing the smooth swimming bacteria data (Fig. 1a) to the run-and-tumble data (Fig. 1b). The orientation fluctuations of tumbling bacteria make it very likely that a bacterium near the SwIM edge at $x=-1$, for example, swims across it, precluding the possibility that it subsequently exits the flow to the right. Hence, we expect to observe much fewer trajectories of run-and-tumble swimmers initially near the left SwIM edge and exiting right relative to smooth swimmers, consistent with the experimental data.

Because the $\dot{\theta}$ equation (7c) is independent of $x$ and $y$, we begin by looking at the effect of noise on the orientation dynamics alone in the hyperbolic flow. The Fokker-Planck equation for the probability density $P(\theta,t)$ restricted to the orientation degree-of-freedom is

$$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial\theta}[-\alpha\sin(2\theta)P]+\frac{\varepsilon}{2}\frac{\partial^{2}P}{\partial\theta^{2}}+\lambda\left(-P+\frac{1}{2\pi}\right).$$

(9)

We focus on the stationary distributions of the $\theta$ variable, which are the stationary solutions ($\partial P/\partial t=0$) of Eq. (9). We first treat the two limiting cases (i) no tumbling ($\lambda=0$), and (ii) no rotational diffusion ($\varepsilon=0$), before proceeding to the case where there is both tumbling and rotational diffusion. Note that the orientation dynamics of a noisy swimmer in the hyperbolic flow is very similar to the orientation dynamics of swimmers in other types external fields, such as magnetotactic swimmers in external magnetic fields Rupprecht2016 ; Waisbord2016 and swimmers in viscocity gradients Stehnach2021 . The main difference here compared to the preceding examples, aside from the source of the torque on the swimmer, is that Eq. (9) is invariant under the symmetry $\theta\mapsto\theta+\pi$.

Here, we present Monte Carlo and semiclassical calculations quantifying the depletion effect. We quantify the depletion effect by calculating the probability $\Pr(x_{0})$ that a swimmer ultimately exits right with a given initial position $-1<x_{0}<1$ and a given intensity of the noise. For an $x_{0}$ near the BIM $x=-1$, the signature of the depletion effect is a decreasing $\Pr(x_{0})$ for increasing noise intensity. A low probability of right-exiting swimmer trajectories initialized near $x=-1$ would be consistent with the absence of such trajectories for run-and-tumble bacteria in the experimental data shown in Fig. 1. Conversely, for an $x_{0}$ near the BIM $x=1$, $\Pr(x_{0})$ should increase with increasing noise intensity. This is simply due to the symmetry of the hyperbolic flow, which requires that

$$\Pr(-x_{0})=1-\Pr(x_{0}).$$

(24)

We focus solely on the dynamics in the $x\theta$ plane, because it is independent of the $y$ variable, as discussed in Sec. III. Hence, Eq. (2) becomes

$$\frac{\partial P}{\partial t}=-\nabla\cdot({\bf f}P)+\frac{\varepsilon}{2}\left(\gamma\frac{\partial^{2}P}{\partial x^{2}}+\frac{\partial^{2}P}{\partial\theta^{2}}\right)+\lambda\left[-P+\frac{1}{2\pi}\int_{0}^{2\pi}P(x,\theta^{\prime},t){\rm d}\theta^{\prime}\right],$$

(25)

where

$${\bf f}=(x+\cos\theta,-\alpha\sin(2\theta))$$

(26)

is the drift restricted to the $x\theta$ plane. In Eq. (25) and throughout this section, we also take $\nabla=(\partial/\partial x,\partial/\partial\theta)$.

We restrict our attention to the case where rotational diffusion dominates translational diffusion, i.e. $\gamma\ll 1$, and we fix $\gamma=0.1$. When $\gamma=0$, all swimmers which cross the line $x=1$ must ultimately exit right, due to the BIM at $x=1$ blocking inward swimming particles. Therefore, the probability to exit right may be calculated by integrating the probability current through $x=1$. We assume this remains approximately true for small $\gamma$. Defining $\Pr(x_{0},t)$ as the probability that a swimmer has exited right by time $t$, we have

$$\Pr(x_{0},t)=\int_{x>1}P(x,\theta,t){\rm d}x{\rm d}\theta.$$

(27)

Differentiating Eq. (27) with respect to time and using Eq. (25), we obtain the probability current

	$\displaystyle\frac{\partial\Pr}{\partial t}$	$\displaystyle=\int_{x>1}\left\{-\nabla\cdot({\bf f}P)+\frac{\varepsilon}{2}\left(\gamma\frac{\partial^{2}P}{\partial x^{2}}+\frac{\partial^{2}P}{\partial\theta^{2}}\right)+\lambda\left[-P+\frac{1}{2\pi}\int_{0}^{2\pi}P(x,\theta^{\prime},t){\rm d}\theta^{\prime}\right]\right\}{\rm d}x{\rm d}\theta$
		$\displaystyle=-\int_{x>1}\nabla\cdot{\bf J}{\rm d}x{\rm d}\theta,$		(28)

where

$${\bf J}=\left[{\bf f}-\frac{\varepsilon}{2}{\mathsf{D}}\nabla(\ln P)\right]P$$

(29)

is the probability current density excluding tumbling, with

$${\mathsf{D}}=\begin{pmatrix}\gamma&0\\ 0&1\end{pmatrix}.$$

(30)

The tumbling contribution in Eq. (28) vanishes upon integration over $\theta$. Using the divergence theorem, the probability current (28) becomes

$$\frac{\partial\Pr}{\partial t}=\int_{0}^{2\pi}{\bf J}(1,\theta,t)\cdot\hat{\bf x}{\rm d}\theta,$$

(31)

and thus the probability to exit right is given by

$$\Pr(x_{0})=\int_{0}^{\infty}{\rm d}t\int_{0}^{2\pi}{\rm d}\theta\left[1+\cos\theta-\frac{\varepsilon\gamma}{2}\frac{\partial}{\partial x}\left(\ln P(1,\theta,t)\right)\right]P(1,\theta,t).$$

(32)

For $\gamma=0$, Eq. (32) is exact. For small $\gamma>0$, Eq. (32) is an approximation, because swimmers close to the righthand side of the BIM may fluctuate over to the lefthand side due to translational diffusion.