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Updated: Jun 14, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
Stochastic process leading to wave equations in dimensions higher than one
1Department of Mathematics, Saint Anselm College, Manchester, New Hampshire 03102, USA. aplyukhin@anselm.edu
Summary
This study introduces novel stochastic processes that mimic classical wave and Klein-Gordon equations. The model uses spatial derivatives for state transitions, enabling wave equation recovery in multiple dimensions.
Area of Science:
- Mathematical Physics
- Computational Physics
Background:
- Classical wave and Klein-Gordon equations describe wave propagation.
- Existing stochastic models like the Goldstein-Kac telegraph process have limitations.
Purpose of the Study:
- To develop novel stochastic processes whose master equations align with classical wave and Klein-Gordon equations.
- To overcome limitations of previous models by incorporating spatial derivatives into state transition dynamics.
Main Methods:
- Developing stochastic processes with master equations.
- Modeling particle motion with constant speed and discrete velocity directions.
- Implementing state transitions dependent on spatial derivatives of population densities.
Main Results:
- Successfully recovered classical wave equations in arbitrary dimensions.
- Demonstrated the ability to imitate Huygens' principle.
- The model deviates from a single-particle description.
Conclusions:
- The proposed stochastic processes offer a new framework for understanding wave phenomena.
- The incorporation of spatial derivatives in transitions is key to recovering wave equations.
- This approach provides a bridge between discrete stochastic dynamics and continuous wave equations.
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