On survival dynamics of classical systems. Non-chaotic open billiards

Vicentini, Eduardo; Kokshenev, V. B.

doi:10.1016/s0378-4371(01)00138-8

Cited by 11 publications

(38 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A polygonal orbit with p vertexes corresponds to (p, 1). A simple star with 5 vertices's is described by (5,2). In an open vesicle, only those periodic orbits with π/p > δ will last, and the peak will develop when n > p. This may be simply understood if we consider that a particle in a periodic orbit characterized by an index p will reach the border of the vesicle at p equally spaced points.…”

Section: Discussion Of the Results And Comparison With Simulationsmentioning

confidence: 99%

“…There is, however, a simplification when δ ≪ 1, that is, if we consider the first terms in the expansion of the distributions in powers of δ. This is the range where other approaches, such as the one in [5], are effective.…”

Section: Discussionmentioning

confidence: 99%

“…We thank Prof. Ronald Dickman for having brought reference [5] to our attention and Profs. Yan Levin, Marco Idiart, and Domingos U. Marchetti for helpful discussions.…”

Section: Acknowledgementsmentioning

confidence: 99%

See 2 more Smart Citations

Particle–wall collision statistics in the open circular billiard

Stilck

2007

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

In the open circular billiard particles are placed initially with a uniform distribution in their positions inside a planar circular vesicle. They all have velocities of the same magnitude, whose initial directions are also uniformly distributed. No particle-particle interactions are included, only specular elastic collisions of the particles with the wall of the vesicle. The particles may escape through an aperture with an angle 2δ. The collisions of the particles with the wall are characterized by the angular position and the angle of incidence. We study the evolution of the system considering the probability distributions of these variables at successive times n the particle reaches the border of the vesicle. These distributions are calculated analytically and measured in numerical simulations. For finite apertures δ < π/2, a particular set of initial conditions exists for which the particles are in periodic orbits and never escape the vesicle. This set is of zero measure, but the selection of angular momenta close to these orbits is observed after some collisions, and thus the distributions of probability have a structure formed by peaks. We calculate the marginal distributions up to n = 4, but for δ > π/2 a solution is found for arbitrary n. The escape probability as a function of n −1 decays with an exponent 4 for δ > π/2 and evidences for a power law decay are found for lower apertures as well.PACS numbers: 45.50.-j,05.20.-y,02.70.-c I. INTRODUCTIONIn the theory of dynamical systems, the study of the decay of simple hamiltonian systems [1]- [5] is of much interest. These studies are an extension of earlier research concerning the closed version of such systems, where the main question is their ergodicity. The pioneering work in this area is centered on the system called Sinai billiard, a circular billiard with a smaller circular exclusion area in its interior [6], which was shown to be ergodic. Other two-dimensional ballistic billiards are known to be ergodic as well, such as the Bunimowitch stadium [4,7]. One main interest in the studies of open billiards is the decay dynamics for long times. Bauer and Bertsch [1] found an exponential decay in an chaotic dynamics and a power law decay for a system with regular dynamics. These first results concerning the integrable case where later questioned by Legrand and Sornette [2], but it became clear that the difficulty in settling this question using numerical experiments is related to the high sensitivity of the results to initial conditions [3]. In a more detailed simulational study for the chaotic two-dimensional Bunimovich stadium [4], algebraic tails were found at sufficiently long times, but the weight of the algebraic tail tends to zero in the limit where the size of the aperture vanishes.In the classical circular billiard the (non-interacting) particles undergo elastic specular collisions with the wall. * Electronic address: jstilck@if.uff.brSince two quantities are conserved (kinetic energy and angular momentum), the system is integrable and therefor...

show abstract

Section: Discussion Of the Results And Comparison With Simulationsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Particle–wall collision statistics in the open circular billiard

Stilck

2007

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

show abstract

“…Such a description of the continuos dynamic system can be derived from the late time behavior of the associated collision subsystem [7,8,10]. The case of m-gon is specified by circumscribing of the rational polygon below a circle of radius r. For a given m, the billiard area A m = (mr 2 /2) sin(2π/m) and the perimeter P m = 2mr sin(π/m) provide the mean collision time…”

Section: Collision Distribution Functionmentioning

confidence: 99%

On Chaotic Dynamics in Rational Polygonal Billiards

Kokshenev¹

2005

SIGMA

View full text Add to dashboard Cite

Abstract. We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular motion in polygons is taken within the alternative deterministic and stochastic frameworks. The analysis is developed in terms of the billiard-wall collision distribution and the particle survival probability, simulated in closed and weakly open polygons, respectively. In the multi-vertex polygons, the late-time wall-collision events result in the circular-like regular periodic trajectories (sliding orbits), which, in the open billiard case are likely transformed into the surviving collective excitations (vortices). Having no topological analogy with the regular orbits in the geometrically corresponding circular billiard, sliding orbits and vortices are well distinguished in the weakly open polygons via the universal and non-universal relaxation dynamics.

show abstract

“…A = T ). Recently, however, it was discovered that a number of decay phenomena observed in nature obeys slower non-exponential relaxation (c.f [2,3,4,5,6])…”

Section: Introduction: Motivation and Aimsmentioning

confidence: 99%

Microscopic explanation of non-Debye relaxation for heat transfer

Fronczak

Hołyst

2007

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

We give a microscopic explanation of both Debye and non-Debye thermalization processes that have been recently reported by Gall and Kutner [1]. Due to reduction of the problem to first passage phenomena we argue that relaxation functions f (t) introduced by the authors directly correspond to survival probabilities S(t) of particles in the considered systems. We show that in the case of broken ergodicity (i.e. in the case of mirror collisions) the survival probability decays as a power law S(t) = τ /t.

show abstract

On survival dynamics of classical systems. Non-chaotic open billiards

Cited by 11 publications

References 23 publications

Particle–wall collision statistics in the open circular billiard

Particle–wall collision statistics in the open circular billiard

On Chaotic Dynamics in Rational Polygonal Billiards

Microscopic explanation of non-Debye relaxation for heat transfer

Contact Info

Product

Resources

About