Statistical properties for an open oval billiard: An investigation of the escaping basins

Hansen, Matheus; Costa, Diogo Ricardo da; Caldas, Iberê L.; Leonel, Edson D.

doi:10.1016/j.chaos.2017.11.036

Cited by 9 publications

(4 citation statements)

References 30 publications

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“…which inserted in the mappings for the average speed and the average quadratic speed (17), results in two coupled difference equations…”

Section: (B) (D))mentioning

confidence: 99%

“…As will be shown later the inhomogeneous nature of F (θ, α) is a fundamental aspect of super diffusion. It can be related to the presence of low period saddles in the static billiard [16], where the collisions occur more often [17], leading to an increase in the distribution value. Figure 4 we show line-integrated profiles of F (α, θ) for both deterministic and random oscillations [14] of the billiard boundary.…”

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Explaining a changeover from normal to super diffusion in time-dependent billiards

Hansen¹,

Ciro²,

Caldas³

et al. 2018

EPL

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The changeover from normal to super diffusion in time dependent billiards is explained analytically. The unlimited energy growth for an ensemble of bouncing particles in time dependent billiards is obtained by means of a two dimensional mapping of the first and second moments of the velocity distribution function. We prove that for low initial velocities the mean velocity of the ensemble grows with exponent ∼ 1/2 of the number of collisions with the border, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent ∼ 1 corresponding to super diffusion. For larger initial velocities, the temporary symmetry in the diffusion of velocities explains an initial plateau of the average velocity. PACS numbers: 05.45.-a, 05.45.Pq, 05.40.FbAs coined by Enrico Fermi [1] Fermi acceleration (FA) is a phenomenon where an ensemble of classical and non interacting particles acquires energy from repeated elastic collisions with a rigid and time varying boundary. It is typically observed in billiards [2-4] whose boundaries are moving in time [5][6][7][8][9]. If the motion of the boundary is random and the initial velocity is small enough [10], the growth of the average velocity is proportional to n 1/2 , with n denoting the number of collisions. If the initial velocity is larger, a plateau of constant velocity is observed in a plot V vs. n which is explained from the symmetry of the velocity diffusion [11]. The symmetry warrants that part of the ensemble grows and part of it decreases in such a way the growing parcel cancels the portion decreasing. As soon as such symmetry is broken the constant regime is changed to a regime of growth. For deterministic oscillations of the border, the scenario is different. Breathing oscillations preserve the shape but not the area of the billiard. It is known that the average velocity evolves in a sub-diffusive manner with a slope of the order of 1/6 [12,13]. For oscillations preserving the area but not the shape of the billiard there are two regimes of growth. For short time the diffusion of velocities is normal passing to super diffusion regime for large enough number of collisions [14]. This changeover is, so far, not yet explained and our contribution in this letter is to fill up this gap in the theory. This is achieved by studying the momenta of the velocity distribution function, noticing that the dynamical angular/time variables have an inhomogeneous distribution in phase space.The results presented in this letter are illustrated by a time-dependent oval-billiard [15] whose phase space is mixed when the boundary is static. The boundary of the billiard is written as R b (θ, t) = 1 + ǫ [1 + a cos(t)] cos(pθ) where R b is the radius of the boundary in polar coordinates, θ is the polar angle, ǫ controls the circle deformation, p > 0 is an integer number [16] given the shape of the boundary, t is the time and a is the amplitude of oscillation of the boundary. Figure 1 shows a typical scenario of the boundary and three collisions illustr...

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“…which inserted in the mappings for the average speed and the average quadratic speed (17), results in two coupled difference equations…”

Section: (B) (D))mentioning

confidence: 99%

mentioning

confidence: 99%

Explaining a changeover from normal to super diffusion in time-dependent billiards

Hansen¹,

Ciro²,

Caldas³

et al. 2018

EPL

Self Cite

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“…The cumbersome structure of the Wada basins implies a particular kind of unpredictability [7], since a small perturbation in the initial conditions lying on a Wada boundary may lead the trajectory to any of the system's attractors. Since the pioneering works of Yorke and collaborators [16,20,21,23], the Wada property has been found in many different cases: chaotic scattering [24,14], Hamiltonian systems [2], fluid dynamics [28], interaction between waves [5], delayed systems [9], black hole shadows [6], etc.…”

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How to detect Wada basins

Wagemakers¹,

Daza²,

Sanjuán³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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“…(3.61). Esse perfil não homogêneo de F (α, θ)é uma característica comum de sistemas mistos [68,69], o que pode sugerir que além da correlação, uma fator importante para a observação da super difusão em bilhares do tipo não breathingé a estrutura mista do espaço de fases.…”

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Propriedades Estatísticas e Termodinâmicas de Bilhares Clássicos

Francisco¹,

Caldas²,

Leonel³

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Statistical properties for an open oval billiard: An investigation of the escaping basins

Cited by 9 publications

References 30 publications

Explaining a changeover from normal to super diffusion in time-dependent billiards

Explaining a changeover from normal to super diffusion in time-dependent billiards

How to detect Wada basins

Propriedades Estatísticas e Termodinâmicas de Bilhares Clássicos

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