Escape of particles in a time-dependent potential well

Costa, Diogo Ricardo da; Dettmann, Carl P.; Leonel, Edson D.

doi:10.1103/physreve.83.066211

Cited by 27 publications

(28 citation statements)

References 35 publications

(38 reference statements)

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“…For our simulations, most of the slower decay was characterized as a power law. Indeed, in the literature it is known that the power law decay, for such cumulative recurrence time distributions for other dynamical systems [24,25], which includes also billiards systems [22,[26][27][28][29], is set in a range of −γ ∈ [1.5,2.5] and that our results match this range. We stress, however, that the total understanding and this behavior is still an open problem, and extensive theoretical and numerical simulations are required to describe its behavior properly.…”

Section: -5supporting

confidence: 75%

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

et al. 2012

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Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables' velocity and time. The system is characterized by a control parameter and experiences a transition from integrable ( = 0) to nonintegrable ( = 0). For small values of , the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter increases and reaches a critical value c , all invariant tori are destroyed and the chaotic sea spreads over the phase space, leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large , the survival probability decays exponentially when it turns into a slower decay as the control parameter is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.

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Section: -5supporting

confidence: 75%

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

et al. 2012

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“…As discussed in Ref. 29, the value of the escape velocity influences the time the particle spends until it reaches such a value but the universal features are unaltered with respect to g. The arrows on right side of each phase space of Fig. 2 represent the values V esc for those values of parameters.…”

Section: Scaling Properties For the Transport On The Chaotic Seamentioning

confidence: 99%

“…31 Indeed, the survival probability was shown to be exponential when the escape from one well to the other one is quick while long trapping leads the survival probability to change from exponential to a power law. For the results, 29 considering large n, depending on the control parameters, the decay curve can exhibit more complicate regimes including stretched exponential. 23 For the range of control parameters considered in the present paper, and for the interval of simulation, the stretched exponential behavior was not observed.…”

Section: -7mentioning

confidence: 99%

“…Recent results discussed for a time-dependent potential well 29 considered the dynamics as a function of three effective control parameters. In our case, keeping fixed the parameter s, we have only the liberty to consider variation of the control parameter e. Therefore, for short n and low energy, if the transport of particles along the chaotic sea was similar to the so called normal Brownian diffusion, the number of entrances of the particles in the region of the active electric field required to diffuse on average until an escaping velocity had to be proportional to V 2 esc .…”

Section: -7mentioning

confidence: 99%

“…There are regions at low energy of phase space where very small islands of regular motion leads the trajectories to experience stickiness from time to time. 29 It was recently studied as an application of a double well potential in chemistry regarding the survival probability and the escape process from one well to the other one due to noise. 31 Indeed, the survival probability was shown to be exponential when the escape from one well to the other one is quick while long trapping leads the survival probability to change from exponential to a power law.…”

Section: -7mentioning

confidence: 99%

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Scaling investigation for the dynamics of charged particles in an electric field accelerator

Ladeira¹,

Leonel

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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. The formalism of escape is used to study the dynamics and hence the transport of charged particles in an accelerator. The model consists of a periodically time dependent electric field limited to a certain region in space that furnishes or absorbs energy of a particle. After the particle leaves the electric field region, a constant magnetic field impels the particle to move in a circular trajectory. This magnetic field is responsible for bringing the particle back to the electric field region leaving the energy unchanged. The control parameters e and s represent the amplitude of the time-dependent electric field and the inverse of the magnetic field, respectively. The parameter e defines the nonlinear strength while ps defines the time that the particle spends in the magnetic region. The dynamics can be described by a two-dimensional nonlinear mapping. Depending on both e and s, the phase space presents either regular or mixed structure. For e ¼ 0, the nonlinear term vanishes and the particle's energy is constant in time. For s ¼ 1; 3; 5; …, there is a strong correlation between the frequency of oscillation of the electric field and the intensity of the magnetic field, and as a consequence, the motion of the particle is regular. For different control parameters than those discussed, the phase space presents mixed structure with islands of regular motion surrounded by chaotic seas and invariant KAM curves that prevent the particle from acquiring unbounded energy gain. We study the escape of particles through a hole in the phase space placed in the velocity axis by considering the position of the lowest energy KAM curve. An ensemb...

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Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges

Knobloch

Krechetnikov

2014

Acta Appl Math

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Escape of particles in a time-dependent potential well

Cited by 27 publications

References 35 publications

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

Scaling investigation for the dynamics of charged particles in an electric field accelerator

Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges

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