Mean first passage time of chiral active Brownian particles

Technical Summary: Mean First Passage Time of Chiral Active Brownian Particles

Problem Statement
Chiral active Brownian particles (CABPs) are self-propelled agents characterized by intrinsic rotational dynamics that induce circular trajectories. While standard active Brownian particles (ABPs) exhibit translational persistence, CABPs break both time-reversal and parity symmetry, fundamentally altering their mechanical response and transport behavior. Understanding how these particles explore confined environments and locate targets is critical for characterizing transport and search efficiency in physical and biological systems, ranging from motile bacteria to synthetic microswimmers. Despite growing interest, a quantitative theory for the mean first passage time (MFPT) of CABPs remains scarce. Existing literature largely focuses on passive or non-chiral active particles, often neglecting intrinsic chirality, or addresses specific narrow escape scenarios without generalizing to diverse geometries. This work addresses the gap in understanding how chirality modulates escape dynamics across various one- and two-dimensional confined domains.

Methodology
The authors employ a theoretical framework based on the backward Fokker–Planck equation to govern the MFPT, $T(x, q)$, defined as the expected time for a particle starting at position $x$ with orientation $q$ to escape a domain. The governing equation is:
$$\mathcal{L}^\dagger[T] = -1$$
where $\mathcal{L}^\dagger$ is the adjoint operator incorporating swim speed ($U_s$), translational diffusivity ($D_x$), angular velocity ($\Omega$), and rotational diffusivity ($D_R$).

The study investigates two canonical geometries in both one (1D) and two (2D) dimensions:

1. 1D Intervals:
   • Symmetric intervals $[-L, L]$ with absorbing boundaries at both ends.
   • Asymmetric intervals $[-L, L]$ with a reflecting boundary on the left and an absorbing boundary on the right.
2. 2D Disks:
   • Circular domains with a single absorbing arc.
   • Circular domains with two symmetric absorbing arcs.

The analysis utilizes a dual approach:

• Asymptotic Analysis: In the high-chirality regime ($\chi \to \infty$, where $\chi = \Omega L / U_s$), the authors employ matched asymptotic expansions. This involves separating the solution into a bulk region and boundary layers near the exits, deriving closed-form asymptotic solutions to capture the scaling of MFPT with chirality.
• Numerical Simulation: For finite chirality and arbitrary parameters, the governing partial differential equations are solved numerically using the Finite Element Method (FEM) via FreeFem++. Results are validated against Brownian Dynamics (BD) simulations for specific cases.

Key Contributions and Results

• High-Chirality Regime (1D):

  • For intervals with two absorbing boundaries, the authors derive a composite asymptotic solution. In the limit of high chirality, the leading-order MFPT converges to that of passive Brownian particles (PBPs).
  • The first correction to the passive result appears at order $O(1/\chi^{3/2})$ for particles starting at the midpoint, while it appears at $O(1/\chi)$ for off-center particles or non-aligned orientations.
  • The analysis reveals that chirality introduces an orientation-dependent correction, creating boundary layers where the particle's rotation steers it away from or toward the exit, leading to non-monotonic dependence on position.

• Finite-Chirality Regime (1D):

  • Two Absorbing Boundaries: For particles initially oriented perpendicular to the exits, the MFPT exhibits non-monotonic behavior as a function of chirality. There exists an optimal intermediate chirality that minimizes the escape time. This optimum arises from a balance where the swim timescale ($L/U_s$) is comparable to the rotational timescale ($1/\Omega$), allowing chirality to steer particles toward the exit before they lose persistence.
  • Reflecting-Absorbing Boundaries: When particles start oriented toward the reflecting wall, chirality significantly enhances escape. For low chirality, particles are trapped by persistent propulsion against the wall; intermediate chirality reorients them away from the wall, reducing the MFPT below that of PBPs. However, for very high chirality, the MFPT increases again as directed motion is suppressed, eventually converging to the PBP limit.

• Two-Dimensional Disk Geometries:

  • Single Absorbing Arc: The MFPT exhibits a sharp minimum at an intermediate chirality value ($\chi \approx 1$) for high Péclet numbers ($Pe$). This optimal chirality is largely independent of the arc size. The non-monotonic behavior is less pronounced at low $Pe$ where diffusion dominates.
  • Two Symmetric Absorbing Arcs: Similar to the 1D symmetric case, non-monotonic behavior is observed for highly active particles ($Pe=10$). However, a key distinction emerges: in the single-arc geometry, CABPs escape slower than PBPs in the limit $\chi \to 0$ (trapped by the reflecting boundary), whereas in the two-arc geometry, CABPs escape faster than PBPs as $\chi \to 0$ because the dual exits allow efficient sliding along the boundary to an exit.

Significance and Claims
The paper establishes that chirality acts as a robust control parameter for transport and search in confined systems. The central finding is the existence of an optimal chirality at an intermediate value that minimizes the MFPT. This phenomenon is particularly pronounced for highly active particles (large Péclet numbers) and in geometries with reflecting boundaries where standard active particles might become trapped.

The authors claim that their work offers a comprehensive characterization of CABP escape dynamics in canonical confinements. They demonstrate that while intermediate chirality enhances escape efficiency by mitigating trapping and steering particles toward targets, excessive chirality is detrimental as it suppresses persistent swimming, causing the system to revert to passive-like diffusion.

The study positions chirality not merely as a biological trait but as a tunable parameter for engineering synthetic active systems. The results suggest that by controlling rotation rates (e.g., via magnetic fields or geometric confinement), one can optimize search strategies for targeted drug delivery, microfluidic transport, and environmental sensing. The work provides a quantitative foundation for understanding how the interplay between chirality and boundary geometry gives rise to non-trivial transport behaviors, extending previous narrow escape theories to broader parameter regimes and geometries.