Abstract
We consider the motion of a particle along the geodesic lines of the Poincaré half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.
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Notes
We omit the calculations for sake of brevity. \(\mathcal {M} _{(q,v)}\) depends on (q, v) as parameters; for simplicity we will use the notation \(\mathcal {M}\) in the following.
Since \(\mathcal {M}\) preserve distances, it sends circumferences into circumferences.
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Acknowledgments
We wish to thank the referees for the appreciation of our work and for the useful remarks. We also thank Dr. Alberto Di Iorio for drawing the pictures of this paper.
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Orsingher, E., Ricciuti, C. & Sisti, F. Motion Among Random Obstacles on a Hyperbolic Space. J Stat Phys 162, 869–886 (2016). https://doi.org/10.1007/s10955-016-1450-y
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DOI: https://doi.org/10.1007/s10955-016-1450-y
Keywords
- Poisson random fields
- Hyperbolic spaces
- Lorentz model
- Boltzmann–Grad limit
- Kinetic equations
- Random flights