Noncrossing run-and-tumble particles on a line.

Area of Science:

• Statistical Physics
• Soft Matter Physics
• Non-equilibrium Systems

Background:

• Active particles exhibit complex dynamics distinct from passive systems.
• Run-and-tumble motion is a key model for self-propelled agents.
• Understanding particle interactions and boundary effects is crucial in non-equilibrium statistical mechanics.

Purpose of the Study:

• To investigate the survival probability of a single run-and-tumble particle near an absorbing wall.
• To exactly compute the non-crossing probability for two independent run-and-tumble particles.
• To analyze the long-time behavior and initial condition dependence of the non-crossing probability.

Main Methods:

• Analytical calculation of survival probability for a single particle with absorbing boundary conditions.
• Exact computation of the probability of no crossing for two independent run-and-tumble particles.
• Analysis of the asymptotic behavior (t^{-1/2} decay) of the non-crossing probability.

Main Results:

• Derived the survival probability of a single run-and-tumble particle at position x and time t.
• Obtained an exact expression for the probability that two independent run-and-tumble particles do not cross.
• Demonstrated that the non-crossing probability decays as t^{-1/2} at large times, with an amplitude dependent on initial conditions, vanishing for negative extrapolation lengths.

Conclusions:

• The problem of two run-and-tumble particles is not reducible to a single particle problem, unlike passive systems.
• The non-crossing probability exhibits universal long-time decay, but its amplitude is sensitive to initial separation.
• The vanishing amplitude at a specific negative extrapolation length is analogous to concepts in neutron transport theory.