One-dimensional telegraphic process with noninstantaneous stochastic resetting

Area of Science:

• Physics
• Statistical Mechanics
• Dynamical Systems

Background:

• Particles undergoing random velocity reversals are common in physical systems.
• Stochastic resetting forces particles back to their origin, altering their dynamics.
• Existing models often assume instantaneous or random-time resets.

Purpose of the Study:

• To investigate the effects of deterministic, position-correlated resetting on particle dynamics.
• To analyze the emergence of stationary states in such systems.
• To determine the first-passage time properties, including mean first-hitting time.

Main Methods:

• Modeling one-dimensional particle motion with constant speed and velocity reversals.
• Implementing a deterministic resetting mechanism where return time depends on current position.
• Deriving analytical formulas for stationary state properties and first-passage times.
• Validating theoretical results with numerical simulations.

Main Results:

• The particle dynamics reach a stationary state under deterministic resetting.
• For specific return dynamics, this stationary state is independent of the reset phase.
• Explicit formulas for the mean first-hitting time were derived.
• Numerical simulations confirmed the analytical findings.

Conclusions:

• Deterministic, position-correlated stochastic resetting leads to predictable stationary states.
• The system's behavior can be understood and quantified through derived first-passage time formulas.
• This work provides a framework for studying resetting processes with non-instantaneous return mechanisms.