Fractional Telegrapher's Equation under Resetting: Non-Equilibrium Stationary States and First-Passage Times

Area of Science:

• Physics
• Applied Mathematics
• Statistical Mechanics

Background:

• Fractional telegrapher's equations model anomalous diffusion processes.
• Stochastic resetting introduces a mechanism to confine systems and influence their long-time behavior.
• Understanding non-equilibrium dynamics is crucial in various physical systems.

Purpose of the Study:

• To analyze two time fractional telegrapher's equations incorporating stochastic resetting.
• To determine the probability density functions and mean squared displacements under these conditions.
• To investigate the first-passage time problem and identify optimal resetting strategies.

Main Methods:

• Integral decomposition method to derive analytical solutions.
• Analysis of long-time limits to understand steady-state behavior.
• Subordination approach using operational time and Lévy stable processes.
• First-passage time analysis to study survival probabilities.

Main Results:

• Probability density functions and mean squared displacements were derived.
• In the long-time limit, systems approach non-equilibrium stationary states.
• Mean squared displacement saturates due to the resetting mechanism.
• The fractional telegraph process was characterized as a subordinated process.
• An optimal resetting rate was found to minimize the mean first-passage time.

Conclusions:

• Stochastic resetting leads to non-equilibrium stationary states and saturation of mean squared displacement.
• The fractional telegraph process can be viewed as a subordinated process.
• Optimal resetting strategies can be identified to control first-passage time properties.