Universal Survival Probability for a d-Dimensional Run-and-Tumble Particle

Area of Science:

• Statistical Mechanics
• Physics
• Complex Systems

Background:

• Active matter systems exhibit unique transport properties.
• Run-and-tumble particles (RTPs) are a fundamental model for active Brownian motion.
• Understanding particle trajectory statistics is crucial in various physical phenomena.

Purpose of the Study:

• To exactly compute the probability S(t) that the x-component of an RTP's position does not change sign up to time t.
• To investigate the dimensionality dependence of this probability for RTPs.
• To explore the universality of this result across different RTP models.

Main Methods:

• Exact computation of the first passage time probability.
• Application of the Sparre Andersen theorem for discrete-time random walks.
• Analysis of RTPs with constant and random tumbling rates and speeds.

Main Results:

• The probability S(t) is independent of the dimensionality (d) for finite times t when tumbling occurs at a constant rate.
• This dimensionality independence is a consequence of the Sparre Andersen theorem.
• The universality extends to RTP models with random speeds drawn from arbitrary distributions.

Conclusions:

• A universal result for the sign-change probability of RTPs is established, independent of system dimensionality and speed distribution.
• The findings highlight the robustness of statistical properties in active matter systems.
• Universality of record statistics in RTP models is demonstrated as a direct consequence.