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Lorentz process with shrinking holes in a wall.
1Department of Stochastics, Mathematical Institute, Budapest University of Technology and Economics, Egry J. u. 1, Budapest 1111, Hungary. nandori@math.bme.hu
Chaos (Woodbury, N.Y.)
|July 5, 2012
Summary
We study a periodic Lorentz process near an almost reflecting wall, finding its limit is a quasi-reflected Brownian motion. This process is Markovian but not strong Markovian, with applications in statistical physics.
Area of Science:
- Probability theory
- Statistical mechanics
- Stochastic processes
Background:
- The periodic Lorentz process models particle motion in a gas.
- Understanding boundary behavior is crucial for diffusion processes.
- Existing models often assume perfectly reflecting or absorbing boundaries.
Purpose of the Study:
- To determine the limiting behavior of a periodic Lorentz process near a time-dependent, partially open boundary.
- To characterize the resulting limiting stochastic process.
- To investigate the properties of local time for this process.
Main Methods:
- Diffusive scaling analysis of the Lorentz process.
- Construction of the limiting process via convergence theorems.
- Analysis of Markovian properties and local time for the limiting process.
Main Results:
- The diffusively scaled limit is identified as a quasi-reflected Brownian motion.
- The limiting process is shown to be Markovian but not strong Markovian.
- New results on the local time of the periodic Lorentz process are established.
Conclusions:
- The study provides a rigorous mathematical framework for diffusion with time-dependent boundary conditions.
- The identified quasi-reflected Brownian motion offers a more realistic model for physical systems.
- The local time results have independent theoretical significance.
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