Resetting of underdamped Brownian motion

Area of Science:

• Statistical Physics
• Non-equilibrium Systems
• Stochastic Processes

Background:

• Brownian motion is a fundamental model for random processes.
• Poissonian resetting introduces periodic returns to a specific state, altering system dynamics.
• Understanding stationary states and convergence is crucial for complex systems.

Purpose of the Study:

• To investigate the effects of two distinct Poissonian resetting procedures on underdamped Brownian motion.
• To analyze the stationary state probability density function (PDF) and its singularities.
• To examine the convergence of the PDF to its stationary state using excess kurtosis.

Main Methods:

• Analytical study of underdamped Brownian motion with two full resetting procedures.
• Procedure 1: Resets both position and velocity to zero.
• Procedure 2: Resets displacement to zero, velocity follows a Maxwell distribution.
• Analysis of the marginal probability density function (PDF) in the stationary state.
• Investigation of excess kurtosis over time to study convergence.
• Consideration of two sets of initial conditions for the second resetting procedure.

Main Results:

• Both resetting procedures lead to stationary state PDFs with exponential tails for large displacements.
• Distinct singularities at the PDF mode are observed: p(x)∼|x|^{-1/3} for the first procedure and p(x)∼-ln|x| for the second.
• The excess kurtosis reveals the convergence behavior, showing high sensitivity to initial conditions for short times in the second procedure.

Conclusions:

• The choice of resetting prescription significantly impacts the stationary state properties of underdamped Brownian motion.
• The specific singularities in the PDF are unique signatures of each resetting strategy.
• Excess kurtosis is a sensitive indicator of convergence dynamics and initial condition dependence in resetting systems.