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Kroetz T, Livorati ALP, Leonel ED, Caldas IL. Global ballistic acceleration in a bouncing-ball model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012905. [PMID: 26274245 DOI: 10.1103/physreve.92.012905] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2015] [Indexed: 06/04/2023]
Abstract
The ballistic increase for the velocity of a particle in a bouncing-ball model was investigated. The phenomenon is caused by accelerating structures in phase space known as accelerator modes. They lead to a regular and monotonic increase of the velocity. Here, both regular and ballistic Fermi acceleration coexist in the dynamics, leading the dynamics to two different growth regimes. We characterized deaccelerator modes in the dynamics, corresponding to unstable points in the antisymmetric position of the accelerator modes. In control parameter space, parameter sets for which these accelerations and deaccelerations constitute structures were obtained analytically. Since the mapping is not symplectic, we found fractal basins of influence for acceleration and deacceleration bounded by the stable and unstable manifolds, where the basins affect globally the average velocity of the system.
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Affiliation(s)
- Tiago Kroetz
- Universidade Tecnológica Federal do Paraná, Pato Branco, Paraná, Brazil
| | - André L P Livorati
- Instituto de Física, Universidade de São Paulo, São Paulo, São Paulo, Brazil
- Departamento de Física, Universidade Estadual Paulista, Rio Claro, São Paulo, Brazil
| | - Edson D Leonel
- Departamento de Física, Universidade Estadual Paulista, Rio Claro, São Paulo, Brazil
- Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
| | - Iberê L Caldas
- Instituto de Física, Universidade de São Paulo, São Paulo, São Paulo, Brazil
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Botari T, Leonel ED. One-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012904. [PMID: 23410401 DOI: 10.1103/physreve.87.012904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2012] [Indexed: 06/01/2023]
Abstract
A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass m, confined to bounce elastically between two rigid walls where one is described by a nonlinear van der Pol type oscillator while the other one is fixed, working as a reinjection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional nonlinear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; and (ii) the case where collisions of the particle do affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter (χ) controlling the nonlinearity of the moving wall. For large χ, a diffusion on the velocity is observed leading to the conclusion that Fermi acceleration is taking place. On the other hand, for case (ii), the motion of the moving wall is affected by collisions with the particle. However, due to the properties of the van der Pol oscillator, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicated organization.
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Affiliation(s)
- Tiago Botari
- Departamento de Física-UNESP-Univ Estadual Paulista, Av. 24A 1515, 13506-900 Rio Claro, SP, Brazil
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Gouve A Ladeira D, Leonel ED. Scaling investigation for the dynamics of charged particles in an electric field accelerator. CHAOS (WOODBURY, N.Y.) 2012; 22:043148. [PMID: 23278083 DOI: 10.1063/1.4772997] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Some dynamical properties of an ensemble of trajectories of individual (non-interacting) classical particles of mass m and charge q interacting with a time-dependent electric field and suffering the action of a constant magnetic field are studied. Depending on both the amplitude of oscillation of the electric field and the intensity of the magnetic field, the phase space of the model can either exhibit: (i) regular behavior or (ii) a mixed structure, with periodic islands of regular motion, chaotic seas characterized by positive Lyapunov exponents, and invariant Kolmogorov-Arnold-Moser curves preventing the particle to reach unbounded energy. We define an escape window in the chaotic sea and study the transport properties for chaotic orbits along the phase space by the use of scaling formalism. Our results show that the escape distribution and the survival probability obey homogeneous functions characterized by critical exponents and present universal behavior under appropriate scaling transformations. We show the survival probability decays exponentially for small iterations changing to a slower power law decay for large time, therefore, characterizing clearly the effects of stickiness of the islands and invariant tori. For the range of parameters used, our results show that the crossover from fast to slow decay obeys a power law and the behavior of survival orbits is scaling invariant.
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Affiliation(s)
- Denis Gouve A Ladeira
- Departamento de Física e Matemática, Univ. Federal de São João del-Rei, UFSJ, Rod. MG 443, Km 7, Fazenda do Cadete, 36420-000 Ouro Branco, MG, Brazil
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de Alcantara Bonfim OF. Dynamical properties of an harmonic oscillator impacting a vibrating wall. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:056212. [PMID: 19518541 DOI: 10.1103/physreve.79.056212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2009] [Indexed: 05/27/2023]
Abstract
The dynamics of a spring-mass system under repeated impact with a vibrating wall is investigated using the static wall approximation. The evolution of the harmonic oscillator is described by two coupled difference equations. These equations are solved numerically, and in some cases exact analytical expressions have also been found. For a periodically vibrating wall, Fermi acceleration is only found at resonance. There, the average rebounding velocity increases linearly with the number of collisions. Near resonance, the average rebounding velocity grows initially with the number of collisions and eventually reaches a plateau. In the vicinity of resonance, the motion of the oscillator exhibits scaling properties over a range of frequency ratios. The presence of dissipation at resonance destroys the Fermi-acceleration process and induces scaling behavior similar to that at near resonance. For a moving wall with a random amplitude at collisions, Fermi acceleration is observed independently of the ratio between the wall and oscillator frequencies. In this case the average rebounding velocity grows with the square root of the number of collisions with the wall. Also, in this latter case, dissipation suppresses the Fermi-acceleration mechanism and induces a scaling behavior with the same universality class as that of the dissipative bouncing ball model with random external perturbations.
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Watkins NW, Credgington D, Sanchez R, Rosenberg SJ, Chapman SC. Kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041124. [PMID: 19518190 DOI: 10.1103/physreve.79.041124] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2008] [Revised: 02/11/2009] [Indexed: 05/27/2023]
Abstract
Lévy flights and fractional Brownian motion have become exemplars of the heavy-tailed jumps and long-ranged memory widely seen in physics. Natural time series frequently combine both effects, and linear fractional stable motion (lfsm) is a model process of this type, combining alpha-stable jumps with a memory kernel. In contrast complex physical spatiotemporal diffusion processes where both the above effects compete have for many years been modeled using the fully fractional kinetic equation for the continuous-time random walk (CTRW), with power laws in the probability density functions of both jump size and waiting time. We derive the analogous kinetic equation for lfsm and show that it has a diffusion coefficient with a power law in time rather than having a fractional time derivative like the CTRW. We discuss some preliminary results on the scaling of burst "sizes" and "durations" in lfsm time series, with applications to modeling existing observations in space physics and elsewhere.
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Affiliation(s)
- N W Watkins
- Physical Sciences Division, British Antarctic Survey (NERC), Cambridge CB3 0ET, United Kingdom
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Livorati ALP, Ladeira DG, Leonel ED. Scaling investigation of Fermi acceleration on a dissipative bouncer model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:056205. [PMID: 19113198 DOI: 10.1103/physreve.78.056205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2008] [Indexed: 05/27/2023]
Abstract
The phenomenon of Fermi acceleration is addressed for the problem of a classical and dissipative bouncer model, using a scaling description. The dynamics of the model, in both the complete and simplified versions, is obtained by use of a two-dimensional nonlinear mapping. The dissipation is introduced using a restitution coefficient on the periodically moving wall. Using scaling arguments, we describe the behavior of the average chaotic velocities on the model both as a function of the number of collisions with the moving wall and as a function of the time. We consider variations of the two control parameters; therefore critical exponents are obtained. We show that the formalism can be used to describe the occurrence of a transition from limited to unlimited energy growth as the restitution coefficient approaches unity. The formalism can be used to characterize the same transition in two-dimensional time-varying billiard problems.
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Affiliation(s)
- André Luis Prando Livorati
- Departamento de Estatística, Matemática Aplicada e Computação, IGCE, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, CEP 13506-900, Rio Claro, São Paulo, Brazil
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Lenz F, Diakonos FK, Schmelcher P. Tunable fermi acceleration in the driven elliptical billiard. PHYSICAL REVIEW LETTERS 2008; 100:014103. [PMID: 18232773 DOI: 10.1103/physrevlett.100.014103] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2007] [Indexed: 05/25/2023]
Abstract
We explore the dynamical evolution of an ensemble of noninteracting particles propagating freely in an elliptical billiard with harmonically driven boundaries. The existence of Fermi acceleration is shown thereby refuting the established assumption that smoothly driven billiards whose static counterparts are integrable do not exhibit acceleration dynamics. The underlying mechanism based on intermittent phases of laminar and stochastic behavior of the strongly correlated angular momentum and velocity motion is identified and studied with varying parameters. The diffusion process in velocity space is shown to be anomalous and we find that the corresponding characteristic exponent depends monotonically on the breathing amplitude of the billiard boundaries. Thus it is possible to tune the acceleration law in a straightforwardly controllable manner.
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Affiliation(s)
- F Lenz
- Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany.
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Karlis AK, Papachristou PK, Diakonos FK, Constantoudis V, Schmelcher P. Fermi acceleration in the randomized driven Lorentz gas and the Fermi-Ulam model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:016214. [PMID: 17677550 DOI: 10.1103/physreve.76.016214] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2007] [Revised: 06/15/2007] [Indexed: 05/16/2023]
Abstract
Fermi acceleration of an ensemble of noninteracting particles evolving in a stochastic two-moving wall variant of the Fermi-Ulam model (FUM) and the phase randomized harmonically driven periodic Lorentz gas is investigated. As shown in [A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis, and P. Schmelcher, Phys. Rev. Lett. 97, 194102 (2006)], the static wall approximation, which ignores scatterer displacement upon collision, leads to a substantial underestimation of the mean energy gain per collision. In this paper, we clarify the mechanism leading to the increased acceleration. Furthermore, the recently introduced hopping wall approximation is generalized for application in the randomized driven Lorentz gas. Utilizing the hopping approximation the asymptotic probability distribution function of the particle velocity is derived. Moreover, it is shown that, for harmonic driving, scatterer displacement upon collision increases the acceleration in both the driven Lorentz gas and the FUM by the same amount. On the other hand, the investigation of a randomized FUM, comprising one fixed and one moving wall driven by a sawtooth force function, reveals that the presence of a particular asymmetry of the driving function leads to an increase of acceleration that is different from that gained when symmetrical force functions are considered, for all finite number of collisions. This fact helps open up the prospect of designing accelerator devices by combining driving laws with specific symmetries to acquire a desired acceleration behavior for the ensemble of particles.
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Affiliation(s)
- A K Karlis
- Department of Physics, University of Athens, GR-15771 Athens, Greece.
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Karlis AK, Papachristou PK, Diakonos FK, Constantoudis V, Schmelcher P. Hyperacceleration in a stochastic Fermi-Ulam model. PHYSICAL REVIEW LETTERS 2006; 97:194102. [PMID: 17155634 DOI: 10.1103/physrevlett.97.194102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2006] [Indexed: 05/12/2023]
Abstract
Fermi acceleration in a Fermi-Ulam model, consisting of an ensemble of particles bouncing between two, infinitely heavy, stochastically oscillating hard walls, is investigated. It is shown that the widely used approximation, neglecting the displacement of the walls (static wall approximation), leads to a systematic underestimation of particle acceleration. An improved approximative map is introduced, which takes into account the effect of the wall displacement, and in addition allows the analytical estimation of the long term behavior of the particle mean velocity as well as the corresponding probability distribution, in complete agreement with the numerical results of the exact dynamics. This effect accounting for the increased particle acceleration--Fermi hyperacceleration--is also present in higher-dimensional systems, such as the driven Lorentz gas.
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Affiliation(s)
- A K Karlis
- Department of Physics, University of Athens, GR-15771 Athens, Greece
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Weron A, Burnecki K, Mercik S, Weron K. Complete description of all self-similar models driven by Lévy stable noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:016113. [PMID: 15697664 DOI: 10.1103/physreve.71.016113] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2003] [Revised: 05/06/2004] [Indexed: 05/24/2023]
Abstract
A canonical decomposition of H-self-similar Lévy symmetric alpha-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to the Lévy symmetric alpha-stable motion are shown to be related to the dissipative and conservative parts of the dynamics. This result provides stochastic analysis tools for study the anomalous diffusion phenomena in the Langevin equation framework. For example, a simple computer test for testing the origins of self-similarity is implemented for four real empirical time series recorded from different physical systems: an ionic current flow through a single channel in a biological membrane, an energy of solar flares, a seismic electric signal recorded during seismic Earth activity, and foreign exchange rate daily returns.
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Affiliation(s)
- Aleksander Weron
- Hugo Steinhaus Center, Institute of Mathematics, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland.
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