Thermodynamics of a bouncer model: A simplified one-dimensional gas

Leonel, Edson D.; Livorati, André L. P.

doi:10.1016/j.cnsns.2014.05.023

Cited by 12 publications

(18 citation statements)

References 37 publications

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“…For the nondissipative version, the system basically behaves like the standard map in a local approximation [2,9], where some of the previous findings concerning the ballistic transport and accelerator modes (AMs) in the standard map serve as the motivation background for this paper [29][30][31][32][33][34][35]. Yet, despite the simple dynamics, interesting applications for this system can be found in dynamic stability in human performance [36], vibration waves in a nanometric-sized mechanical contact system [37], granular materials [38], experimental devices concerning normal coefficient of restitution [39], mechanical vibrations [40,41], anomalous transport and diffusion [42], thermodynamics [43], crisis between chaotic attractors [44], and chaos control [45], among others [46,47].…”

Section: Introductionmentioning

confidence: 99%

Transition from normal to ballistic diffusion in a one-dimensional impact system

et al. 2018

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We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.

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Section: Introductionmentioning

confidence: 99%

Transition from normal to ballistic diffusion in a one-dimensional impact system

et al. 2018

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“…The case of a single particle bouncing freely upon a vibrating surface has also attracted much attention, first as an example of nonlinear system exhibiting a transition to chaos by period doubling [17,18]. Authors also considered the statistical properties of the particle in the chaotic regime, which can be viewed as a simplified one-dimensional gas [19] or a dissipative system maintained in a permanent regime [20].…”

Section: Introductionmentioning

confidence: 99%

Transversal stability of the bouncing ball on a concave surface

et al. 2015

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A ball bouncing repeatedly on a vertically vibrating surface constitutes the famous "bouncing ball" problem, a nonlinear system used in the 1980s, and still in use nowadays, to illustrate the route to chaos by period doubling. In experiments, in order to avoid the ball escape that would be inevitable with a flat surface, a concave lens is often used to limit the horizontal motion. However, we observe experimentally that the system is not stable. The ball departs from the system axis and exhibits a pendular motion in the permanent regime. We propose theoretical arguments to account for the decrease of the growth rate and of the asymptotic-size of the trajectory when the frequency of the vibration is increased. The instability is very sensitive to the physics of the contacts, which makes it a potentially interesting way to study the collisions rules, or to test the laws used in numerical studies of granular matter.

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“…So, taking the expression of (∆V ) 2 in the interval between collisions, we may interpret this interval as an integration variable [48,49], where one may set that…”

Section: A Root Mean Square Velocity (Vrms)mentioning

confidence: 99%

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

Livorati

Dettmann

Caldas

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The transport and diffusion properties for the velocity of a Fermi-Ulam model were characterized using the decay rate of the survival probability. The system consists of an ensemble of non interacting particles confined to move along and experience elastic collisions with two infinitely heavy walls. One is fixed, working as a returning mechanism of the colliding particles while the other one moves periodically in time. The diffusion equation is solved and the diffusion coefficient is numerically estimated by means of the averaged square velocity. Our results show remarkably good agreement of the theory and simulation for the chaotic sea below the first elliptic island in the phase space. From the decay rates of the survival probability, we obtained transport properties that can be extended to other nonlinear mappings, as well to billiard problems. We study the dynamics of an ensemble of non interacting particles moving constrained by two infinitely heavy walls, where one of then is moving periodically in time and the other is fixed. This problem, also known as Fermi-Ulam model, has application in many areas, including astrophysics, atom-optics, quantum mechanics, among others. The diffusive behaviour of the velocity, here set as the way the transport of orbits occurs in the phase space, is investigated considering transport properties obtained from the decay rate of the survival probability, defined by means of escape formalism. Since the system present mixed dynamics, stickiness phenomenon may influence the transport causing anomalous diffusion. In this study we developed an analytical approach for the diffusion coefficient along the transport through the chaotic sea considering escape rate formalism and survival probability analysis. The numerical results we obtained are in good agreement with the theory, and confirm the robustness of the formalism. The results obtained here can be extended to other similar dynamical systems.

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Thermodynamics of a bouncer model: A simplified one-dimensional gas

Cited by 12 publications

References 37 publications

Transition from normal to ballistic diffusion in a one-dimensional impact system

Transition from normal to ballistic diffusion in a one-dimensional impact system

Transversal stability of the bouncing ball on a concave surface

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

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