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Drifting diffusion on a circle as continuous limit of a multiurn Ehrenfest model
1Institute of Optical Sciences, National Central University, Chung-Li 32054, Taiwan, Republic of China.
Summary
We analyze the continuous version of the Ehrenfest urn model, revealing a diffusion process on a circle. This study details short and long-term behaviors, connecting it to quantum mechanics.
Area of Science:
- Statistical Mechanics
- Mathematical Physics
- Diffusion Processes
Background:
- The Ehrenfest urn model describes particle distribution between two boxes.
- Previous work established a discrete multibox model.
Purpose of the Study:
- To investigate the continuous limit of the multibox Ehrenfest urn model.
- To analyze the resulting diffusion process and its properties.
- To explore connections with quantum mechanics.
Main Methods:
- Deriving a governing differential equation for the continuous system.
- Solving the differential equation to determine short-time behavior.
- Applying the Poisson summation formula for long-time behavior analysis.
Main Results:
- The continuous system exhibits diffusion on a circle with drift.
- Explicit solutions for short-time and long-time behaviors were obtained.
- Results were validated against previous findings in the large M limit.
Conclusions:
- The continuous limit provides a tractable model for diffusion processes.
- The study establishes a link between this diffusion model and Schrödinger equations in quantum mechanics.