Most probable path of an active Brownian particle

TL;DR

This paper derives and analyzes the most probable transition paths of active Brownian particles in two dimensions, revealing shape transitions and confirming findings through simulations, advancing understanding of rare events in active matter.

Contribution

It introduces explicit solutions for the most probable paths of active Brownian particles and demonstrates their nonuniqueness and shape transitions using the Onsager--Machlup variational principle.

Findings

Most probable path shape transitions from I to U to L with increasing transition time.

Numerical and theoretical analyses agree on path shape changes.

Simulation confirms the shape transition phenomena.

Abstract

In this study, we investigate the transition path of a free active Brownian particle (ABP) on a two-dimensional plane between two given states. The extremum conditions for the most probable path connecting the two states are derived using the Onsager--Machlup integral and its variational principle. We provide explicit solutions to these extremum conditions and demonstrate their nonuniqueness through an analogy with the pendulum equation indicating possible multiple paths. The pendulum analogy is also employed to characterize the shape of the globally most probable path obtained by explicitly calculating the path probability for multiple solutions. We comprehensively examine a translation process of an ABP to the front as a prototypical example. Interestingly, the numerical and theoretical analyses reveal that the shape of the most probable path changes from an I to a U shape and to the $\ell$ shape with an increase in the transition process time. The Langevin simulation also confirms this shape transition. We also discuss further method applications for evaluating a transition path in rare events in active matter.