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Simple jumping process with memory: transport equation and diffusion.

A Kamińska1, T Srokowski

  • 1Institute of Nuclear Physics, PL-31-342 Kraków, Poland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 13, 2004
PubMed
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We introduce a generalized kangaroo process, a stochastic jumping model that remembers past states. This novel approach models systems relaxing to non-Maxwellian distributions and includes diffusion as a special case.

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.

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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.