A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincaré half-plane and Poincaré disk) is examined. Each particle can split into two particles only once at Poisson spaced times and deviates orthogonally when splitted. At time t , after N (t ) Poisson events, there are N (t ) + 1 particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c (hyperbolic velocity of motion) and λ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented.

Cascades of particles moving at finite velocity in hyperbolic spaces / Cammarota, Valentina; Orsingher, Enzo. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 133:(2008), pp. 1137-1159. [10.1007/s10955-008-9648-2]

Cascades of particles moving at finite velocity in hyperbolic spaces

CAMMAROTA, VALENTINA
Membro del Collaboration Group
;
ORSINGHER, Enzo
Membro del Collaboration Group
2008

Abstract

A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincaré half-plane and Poincaré disk) is examined. Each particle can split into two particles only once at Poisson spaced times and deviates orthogonally when splitted. At time t , after N (t ) Poisson events, there are N (t ) + 1 particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c (hyperbolic velocity of motion) and λ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented.
2008
branching processes; difference-differential equations; hyperbolic brownian motion; hyperbolic trigonometry; laplace transforms; non-euclidean geometry; random motions
01 Pubblicazione su rivista::01a Articolo in rivista
Cascades of particles moving at finite velocity in hyperbolic spaces / Cammarota, Valentina; Orsingher, Enzo. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - 133:(2008), pp. 1137-1159. [10.1007/s10955-008-9648-2]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/360336
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