Intermittent Kac's flights and the generalized telegrapher's equation

Area of Science:

• Statistical Physics
• Stochastic Processes
• Mathematical Physics

Background:

• The study addresses random differential equations, specifically those involving intermittent velocity changes in Kac's flight.
• Existing models often simplify velocity dynamics, necessitating more generalized approaches.

Purpose of the Study:

• To propose and solve a generalized one-dimensional telegrapher equation for intermittent velocity changes.
• To analyze the resulting finite-velocity diffusion-like process and its statistical properties.

Main Methods:

• Utilized the enlarged master equation approach to derive exact equations for distribution evolution.
• Investigated the second moment of the profile evolution under non-Poisson statistics.
• Presented numerical simulations for various initial profiles.

Main Results:

• Obtained an exact differential equation for the normalized positive distribution.
• Characterized the ballistic regime, its cutoff, and time-dependent Gaussian convergence.
• Analyzed the influence of non-Poisson statistics on the second moment.

Conclusions:

• The proposed model provides a framework for understanding generalized random flights with intermittent stochastic velocity.
• The study offers insights into diffusion processes with non-standard waiting time distributions.
• The findings are relevant for systems exhibiting complex random dynamics.