Simple jumping process with memory: transport equation and diffusion

Area of Science:

• Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• The kangaroo process is a known model for stochastic systems.
• Existing models often lack memory of previous states.
• Modeling systems relaxing to non-Maxwellian distributions is challenging.

Purpose of the Study:

• To present a generalized stochastic jumping process.
• To incorporate memory of previous states into the process.
• To model systems relaxing to non-Maxwellian distributions.

Main Methods:

• Defining a stochastic jumping process with jump-size probability density and jumping rate.
• Considering both pre- and post-jump process values to preserve memory.
• Interpreting the master equation as a kinetic equation with variable collision rate.

Main Results:

• The generalized process preserves memory of previous values.
• The process exhibits a simple stationary limit.
• It can model systems relaxing to distributions other than Maxwellian.
• Constant jumping rate leads to normal or ballistic diffusion.

Conclusions:

• The presented stochastic jumping process offers a flexible framework for modeling complex systems.
• Its ability to retain memory makes it suitable for systems with history-dependent dynamics.
• The model provides a unified view encompassing diffusion and more complex relaxation behaviors.