Run-and-tumble particle with saturating rates

Area of Science:

• Statistical Physics
• Non-equilibrium Systems
• Stochastic Processes

Background:

• Run-and-tumble particle models are crucial for understanding biological and chemical transport.
• Previous models often assumed unbounded rates, limiting real-world applicability.
• Space-dependent rates introduce complex dynamics not fully explored.

Purpose of the Study:

• To investigate the behavior of a run-and-tumble particle with space-dependent, saturating rates.
• To analyze the steady-state distribution and mean-squared displacement under these conditions.
• To compare findings with existing models and provide quantitative insights.

Main Methods:

• Analytical treatment of a one-dimensional run-and-tumble model.
• Derivation of the steady-state distribution for saturating rate functions.
• Calculation of the mean-squared displacement.
• Exact solution for a simplified step-function rate case.

Main Results:

• Existence of a stationary state with exponentially decaying or faster steady-state distributions.
• Unimodal or bimodal distribution shapes observed.
• Non-monotonic or plateauing mean-squared displacement for particles starting far from the origin.
• Consistency between numerical results and the exact solution of a limiting case.

Conclusions:

• Saturating rates at large distances lead to distinct particle dynamics compared to unbounded rate models.
• The boundedness of rates significantly impacts the approach to the stationary state.
• The study provides a more physically realistic framework for run-and-tumble particle modeling.