Unlimited Energy Growth Research Articles

Characterizing a transition from limited to unlimited diffusion in energy for a time-dependent stochastic billiard.

We explore Fermi acceleration in a stochastic oval billiard which shows unlimited to limited diffusion in energy when passing from the free to the dissipative case. We provide evidence for a transition from limited to unlimited energy growth taking place while detuning the corresponding restitution coefficient responsible for the degree of dissipation. A corresponding order parameter is suggested, and its susceptibility is shown to diverge at the critical point. We show that this order parameter is also be applicable to the periodically driven oval billiard and discuss the elementary excitation of the controlled diffusion process.

Nature, Nation and the Dam. Narratives about the Harnessed Waterfall in Early Twentieth-Century Sweden

When the first large-scale hydroelectric power plants were built in Sweden at the beginning of the twentieth century, the waterfall became a twofold symbol of the Swedish nation. On the one hand, the harnessed waterfall promised unlimited energy and economic growth, and thus turned into a unifying symbol of a national productive landscape. On the other hand, the unharnessed wild waterfall, to be enjoyed by tourists seeking refuge from the modern industrial world, became a symbol of a national recreational landscape. This situation, however problematic, did not result in conflict. Taking a look at the first two state-built water power plants in Sweden, I trace how the two contrasting concepts of landscape were harmonized within public discourse. I demonstrate how engineers, architects, conservationists, tourist organisations, and journalists together produced a broad public acceptance of the drastic changes in the national landscape brought about by the construction of these power stations.

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

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 speed distribution function. We prove that, for low initial speeds the average speed of the ensemble grows with exponent of the number of collisions with the boundary, therefore exhibiting normal diffusion. Eventually, this regime changes to a faster growth characterized by an exponent corresponding to super diffusion. For larger initial energies, the temporary symmetry in the diffusion of speeds explains an initial plateau of the average speed.

Limited fetch revisited: Comparison of wind input terms, in surface wave modeling

Results pertaining to numerical solutions of the Hasselmann kinetic equation (HE), for wind driven sea spectra, in the fetch limited geometry, are presented. Five versions of source functions, including the recently introduced ZRP model (Zakharov et al., 2012), have been studied, for the exact expression of Snl and high-frequency implicit dissipation, due to wave-breaking. Four of the five experiments were done in the absence of spectral peak dissipation for various Sin terms. They demonstrated the dominance of quadruplet wave–wave interaction, in the energy balance, and the formation of self-similar regimes, of unlimited wave energy growth, along the fetch. Between them was the ZRP model, which strongly agreed with dozens of field observations performed in the seas and lakes, since 1947. The fifth, the WAM3 wind input term experiment, used additional spectral peak dissipation and reproduced the results of a previous, similar, numerical simulation described in Komen et al. (1994), but only supported the field experiments for moderate fetches, demonstrating a total energy saturation at half of that of the Pierson–Moscowits limit. The alternative framework for HE numerical simulation is proposed, along with a set of tests, allowing one to select physically-justified source terms.

Pinball Dynamics: Unlimited Energy Growth in Switching Hamiltonian Systems

A family of discontinuous symplectic maps arising naturally in the study of nonsmooth switched Hamiltonian systems is considered. This family depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving transformations due to nonlinear resonances. This paper provides a general description of the map and a construction of nontrivial unbounded solutions for the special case of the pinball transformation. An asymptotic expansion of the pinball map in the limit of large energy is derived and used for the construction of unbounded solutions. For the generic values of the parameters, in the large energy limit, the map behaves similarly to another one considered earlier by Kesten (Acta Arith 1966).

Dynamics of a charged particle in a dissipative Fermi–Ulam model

The dynamics of a metallic particle confined between charged walls is studied. One wall is fixed and the other moves smoothly and periodically in time. Dissipation is considered by assuming a friction produced by the contact between the particle and a rough surface. We investigate the phase space of the simplified and complete versions of the model. Our results include (i) coexistence of islands of regular motion with an attractor located at the low energy portion of phase space in the complete model and; (ii) coexistence of attractors with trajectories that present unlimited energy growth in the simplified model.

Dynamical and statistical properties of a rotating oval billiard

Some dynamical and statistical properties of a time-dependent rotating oval billiard are studied. We considered cases with (i) positive and (ii) negative curvature for the boundary. For (i) we show the system does not present unlimited energy growth. For case (ii) however the average velocity for an ensemble of noninteracting particles grows as a power law with acceleration exponent well defined. Finally, we show for both cases that after introducing time-dependent perturbation, the mixed structure of the phase space observed for static case is recovered by making a suitable transformation in the angular position of the particle.

Competition between unlimited and limited energy growth in a two-dimensional time-dependent billiard

Abstract Some dynamical properties for a dissipative time-dependent Lorentz gas are studied. We assume that the size of the scatterers change periodically in time. We show that for some combination of the control parameters the particles come to a complete stop between the scatterers, but for some other cases, the average velocity grows unbounded. This is the first time that the unlimited energy growth is observed in a dissipative system. Finally, we study the behavior of the average velocity as a function of the number of collisions and we show that the system is scaling invariant with scaling exponents well defined.

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

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 ε and experiences a transition from integrable (ε=0) to nonintegrable (ε≠0). For small values of ε, the phase space shows a mixed structure where periodic islands, chaotic seas, and invariant tori coexist. As the parameter ε increases and reaches a critical value εc, 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 ε, the survival probability decays exponentially when it turns into a slower decay as the control parameter ε is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity.

SCALING INVARIANCE IN A TIME-DEPENDENT ELLIPTICAL BILLIARD

We study some dynamical properties of a classical time-dependent elliptical billiard. We consider periodically moving boundary and collisions between the particle and the boundary are assumed to be elastic. Our results confirm that although the static elliptical billiard is an integrable system, after introducing time-dependent perturbation on the boundary the unlimited energy growth is observed. The behavior of the average velocity is described using scaling arguments.

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Some dynamical properties for a time dependent Lorentz gas considering both the dissipative and non dissipative dynamics are studied. The model is described by using a four-dimensional nonlinear mapping. For the conservative dynamics, scaling laws are obtained for the behavior of the average velocity for an ensemble of non interacting particles and the unlimited energy growth is confirmed. For the dissipative case, four different kinds of damping forces are considered namely: (i) restitution coefficient which makes the particle experiences a loss of energy upon collisions; and in-flight dissipation given by (ii) F=-ηV(2); (iii) F=-ηV(μ) with μ≠1 and μ≠2 and; (iv) F=-ηV, where η is the dissipation parameter. Extensive numerical simulations were made and our results confirm that the unlimited energy growth, observed for the conservative dynamics, is suppressed for the dissipative case. The behaviour of the average velocity is described using scaling arguments and classes of universalities are defined.

Decay of energy and suppression of Fermi acceleration in a dissipative driven stadium-like billiard

The behavior of the average energy for an ensemble of non-interacting particles is studied using scaling arguments in a dissipative time-dependent stadium-like billiard. The dynamics of the system is described by a four dimensional nonlinear mapping. The dissipation is introduced via inelastic collisions between the particles and the moving boundary. For different combinations of initial velocities and damping coefficients, the long time dynamics of the particles leads them to reach different states of final energy and to visit different attractors, which change as the dissipation is varied. The decay of the average energy of the particles, which is observed for a large range of restitution coefficients and different initial velocities, is described using scaling arguments. Since this system exhibits unlimited energy growth in the absence of dissipation, our results for the dissipative case give support to the principle that Fermi acceleration seems not to be a robust phenomenon.

A peculiar Maxwell’s Demon observed in a time-dependent stadium-like billiard

The dynamics of a driven stadium-like billiard is considered using the formalism of discrete mappings. The model presents a resonant velocity that depends on the rotation number around fixed points and external boundary perturbation which plays an important separation rule in the model. We show that particles exhibiting Fermi acceleration (initial velocity is above the resonant one) are scaling invariant with respect to the initial velocity and external perturbation. However, initial velocities below the resonant one lead the particles to decelerate therefore unlimited energy growth is not observed. This phenomenon may be interpreted as a specific Maxwell’s Demon which may separate fast and slow billiard particles.

Scaling invariance for the escape of particles from a periodically corrugated waveguide

The escape dynamics of a classical light ray inside a corrugated waveguide is characterised by the use of scaling arguments. The model is described via a two-dimensional nonlinear and area preserving mapping. The phase space of the mapping contains a set of periodic islands surrounded by a large chaotic sea that is confined by a set of invariant tori. When a hole is introduced in the chaotic sea, letting the ray escape, the histogram of frequency of the number of escaping particles exhibits rapid growth, reaching a maximum value at np and later decaying asymptotically to zero. The behaviour of the histogram of escape frequency is characterised using scaling arguments. The scaling formalism is widely applicable to critical phenomena and useful in characterisation of phase transitions, including transitions from limited to unlimited energy growth in two-dimensional time varying billiard problems.

Fermi acceleration and its suppression in a time-dependent Lorentz gas

Some dynamical properties for a Lorentz gas were studied considering both static and time-dependent boundaries. For the static case, it was confirmed that the system has a chaotic component characterized with a positive Lyapunov exponent. For the time-dependent perturbation, the model was described using a four-dimensional nonlinear map. The behaviour of the average velocity is considered in two different situations: (i) non-dissipative and (ii) dissipative dynamics. Our results confirm that unlimited energy growth is observed for the non-dissipative case. However, and totally new for this model, when dissipation via inelastic collisions is introduced, the scenario changes and the unlimited energy growth is suppressed, thus leading to a phase transition from unlimited to limited energy growth. The behaviour of the average velocity is described using scaling arguments.

Suppressing Fermi acceleration in two-dimensional driven billiards

We consider a dissipative oval-like shaped billiard with a periodically moving boundary. The dissipation considered is proportional to a power of the velocity V of the particle. The three specific types of power laws used are: (i) F∝-V ; (ii) F∝-V(2) and (iii) F∝-V(δ) with 1<δ<2 . In the course of the dynamics of the particle, if a large initial velocity is considered, case (i) shows that the decay of the particle's velocity is a linear function of the number of collisions with the boundary. For case (ii), an exponential decay is observed, and for 1<δ<2 , an powerlike decay is observed. Scaling laws were used to characterize a phase transition from limited to unlimited energy gain for cases (ii) and (iii). The critical exponents obtained for the phase transition in the case (ii) are the same as those obtained for the dissipative bouncer model. Therefore near this phase transition, these two rather different models belong to the same class of universality. For all types of dissipation, the results obtained allow us to conclude that suppression of the unlimited energy growth is indeed observed.

Suppressing Fermi Acceleration in a Driven Elliptical Billiard

We study dynamical properties of an ensemble of noninteracting particles in a time-dependent elliptical-like billiard. It was recently shown [Phys. Rev. Lett. 100, 014103 (2008)] that for the nondissipative dynamics, the particle experiences unlimited energy growth. Here we show that inelastic collisions suppress Fermi acceleration in a driven elliptical-like billiard. This suppression is yet another indication that Fermi acceleration is not a structurally stable phenomenon.

Competition between suppression and production of Fermi acceleration

The behavior of the average velocity for a classical particle in the one-dimensional Fermi accelerator model under sawtooth external force is considered. For elastic collisions, it is known that the average velocity of the particle grows unlimitedly because of the discontinuities of the derivative of the moving wall's position with respect to time. However, and contrary to what was expected to be observed, the introduction of a friction force generated from a slip of a body against a rough surface leads to a boundary separating different regions of the phase space that yields the particle to either experience unlimited energy growth or suppression of Fermi acceleration. The Fermi acceleration is described by using scaling arguments. The formalism presented can be extended to two-dimensional time-dependent billiards as well as to higher-order mappings.

Time‐Dependent Billiards

We discuss in this chapter some dynamical properties for time dependent billiards. We construct the equations of the mapping that describe the dynamics of the particles considering that the velocity of the particle is given by the application of the momentum conservation law at each impact with the moving boundary. After the collision, the velocity of the particle changes, consequently a new pair of variables is added to the usual pair of angular variables, namely the velocity of the particle after the collision and the instant of the collision. We discuss the Loskutov–Ryabov–Akinshin (LRA) conjecture that says the existence of chaos in the billiard with static boundary is sufficient condition for the Fermi acceleration (unlimited energy growth) when the particle is time dependent. The conjecture was tested in the oval billiard leading to the unlimited energy growth. In the elliptical billiard, which is integrable for the fixed boundary, the time dependency of the boundary transforms the separatrix curve into a distribution of points called as stochastic layer hence leading to the unlimited energy growth and producing the Fermi acceleration.

Scaling investigation of Fermi acceleration on a dissipative bouncer model

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.