Microorganism billiards in closed plane curves

Abstract

Recent experiments have shown that many species of microorganisms leave a solid surface at a fixed angle determined by steric interactions and near-field hydrodynamics. This angle is completely independent of the incoming angle. For several collisions in a closed body this determines a unique type of billiard system, an aspecular billiard in which the outgoing angle is fixed for all collisions. We analyze such a system using numerical simulation of this billiard for varying tables and outgoing angles, and also utilize the theory of one-dimensional maps and wavefront dynamics. When applicable we cite results from and compare our system to similar billiard systems in the literature. We focus on examples from three broad classes: the ellipse, the Bunimovich billiards, and the Sinai billiards. The effect of a noisy outgoing angle is also discussed.

Rebound and scattering of motile Chlamydomonas algae in confined chambers

“…Microorganisms billiards with constant scattering angle θ out , also motivated by experimental work, 41 have been previously studied in polygonal geometries 36 as well as ellipses and more complex closed curves. 52 We extend here these analysis to the particular geometry of the Plateau border section, thus providing a link between past work on microorganisms billiards and our new experiments.…”

“…This choice for θ out is an important difference with well-studied classical billiards 49 , including three-disks ones 50,51 , where the particle reflection is specular, meaning that θ in = θ out . Microorganisms billiards with constant scattering angle θ out , also motivated by experimental work 41 , have been previously studied in polygonal geometries 36 as well as ellipses and more complex closed curves 52 . We extend here these analysis to the particular geometry of the Plateau border section, thus providing a link between past work on microorganisms billiards and our new experiments.…”

Rebound and scattering of motile Chlamydomonas algae in confined chambers

“…This choice for θ out is an important difference with well-studied classical billiards [12], including three-disks ones [43,56], where the particle reflection is specular, meaning that θ in = θ out . Microorganisms billiards with constant scattering angle θ out , also motivated by experimental work [28], have been previously studied in polygonal geometries [49] as well as ellipses and more complex closed curves [29]. We extend here these analysis to the particular geometry of the Plateau border section, thus providing a link between past work on microorganisms billiards and our new experiments.…”

Microbial narrow-escape is facilitated by wall interactions

“…This beautiful and mathematically intricate problem, first studied by Lord Rayleigh in the context of acoustics [18], has recently received much attention as the mean escape time controls the rates of fundamental molecular processes (e.g., from mRNA escaping through nucleus pores in the cell [19] to signalling in dendritic spines [20,21]). Instead, for non-Brownian particles following purely ballistic motion, the escape dynamics is captured by the theory of leaking chaotic systems [22], with an exponential decay in particle number only expected for chaotic dynamics, while so-called deterministic 'billiards' give rise to a 1/t decay [23][24][25].…”