Motion Among Random Obstacles on a Hyperbolic Space
TLDR
In this article, the authors consider the case where a particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless, and they show that their process converges to a Markovian process, namely a random flight on the hyperbolic manifold.Abstract:
We consider the motion of a particle along the geodesic lines of the Poincare 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.read more
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A Complete Bibliography of the Journal of Statistical Physics: 2000{2009
TL;DR: In this paper, Zuc11b et al. this paper showed that 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15], 1/n [Per17] and 1/m [DFL17] were the most frequent p ≤ p ≥ ∞.
References
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Book
Statistical Mechanics: A Short Treatise
TL;DR: In this paper, the authors propose a statistical ensembles and a phase transition model for statistical mechanics, which is based on Brownian motion and coarse graining and nonequilibrium.
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Markov Processes, Semigroups and Generators
TL;DR: Brownian Motion, Markov Processes, Martingales, and Markov Semigroups have been studied extensively in the literature as discussed by the authors, with a focus on probability and analysis.
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On the Boltzmann equation for the Lorentz gas
TL;DR: In this paper, the Boltzmann-Grad limit for the wind-tree model was considered and it was shown that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c.
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The Lorentz process converges to a random flight process
TL;DR: In this article, the Boltzmann-Grad limit is considered and it is shown that the Lorentz process converges in the weak*-topology of regular Borel measures on the paths space to some stochastic process.
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Isotropic transport process on a Riemannian manifold
TL;DR: In this paper, a canonical Markov process on the tangent bundle of a complete Riemannian manifold was constructed, which generalizes the isotropic scattering transport process on Euclidean space.