Semiclassical technique for computing noisy swimmer escape probabilities in fluid flows

The random fluctuations inherent to the self-propulsion of microswimmers can have a dramatic impact on their trajectories, especially near flow-induced transport barriers. We discuss a method for calculating the probability of particular noisy swimmer trajectories in a given fluid flow, taking the hyperbolic flow as a case study. Our approach consists of computing solutions to the time-dependent Fokker-Planck equation of a swimmer in the weak-noise limit. 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. In contrast, we construct a time-dependent swimmer probability density function by following the Lagrangian paths of a swimmer. This procedure mirrors the semiclassical approximation in quantum mechanics and similarly involves calculating the least-action paths of a Hamiltonian system derived from the swimmer's Fokker-Planck equation. We use this approach to quantify the tendency of noisy swimmers to cross a transport barrier in the hyperbolic flow and compare our results with Monte Carlo calculations.