Most probable path of an active Brownian particle

Area of Science:

• Physics
• Statistical Mechanics
• Soft Matter Physics

Background:

• Active Brownian particles (ABPs) are fundamental models for self-propelled entities.
• Understanding transition paths is crucial for predicting the behavior of active matter.
• The Onsager-Machlup theory provides a framework for analyzing most probable paths in stochastic systems.

Purpose of the Study:

• To investigate the transition path of a free active Brownian particle (ABP) between two states on a 2D plane.
• To derive and solve the extremum conditions for the most probable path.
• To analyze the shape transitions of these paths and their dependence on time.

Main Methods:

• Utilizing the Onsager-Machlup integral and its variational principle to derive extremum conditions.
• Employing an analogy with the pendulum equation to demonstrate path nonuniqueness and characterize path shapes.
• Performing numerical and theoretical analyses, including Langevin simulations, for a translation process example.

Main Results:

• Explicit solutions for extremum conditions were found, revealing nonunique paths.
• The shape of the most probable path transitions from I to U to ℓ shapes as transition time increases.
• Langevin simulations confirmed these path shape transitions.

Conclusions:

• The study provides a method for evaluating transition paths in active matter, particularly for rare events.
• The nonuniqueness of paths and their dynamic shape evolution are key findings.
• This work offers a deeper understanding of stochastic processes in active systems.