Time-dependent Billiard 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.

Dynamical thermalization in time-dependent billiards.

Numerical experiments of the statistical evolution of an ensemble of noninteracting particles in a time-dependent billiard with inelastic collisions reveals the existence of three statistical regimes for the evolution of the speed ensemble, namely, diffusion plateau, normal growth/exponential decay, and stagnation. These regimes are linked numerically to the transition from Gauss-like to Boltzmann-like speed distributions. Furthermore, the different evolution regimes are obtained analytically through velocity-space diffusion analysis. From these calculations, the asymptotic root mean square of speed, initial plateau, and the growth/decay rates for an intermediate number of collisions are determined in terms of the system parameters. The analytical calculations match the numerical experiments and point to a dynamical mechanism for "thermalization," where inelastic collisions and a high-dimensional phase space lead to a bounded diffusion in the velocity space toward a stationary distribution function with a kind of "reservoir temperature" determined by the boundary oscillation amplitude and the restitution coefficient.

Ensemble separation and stickiness influence in a driven stadium-like billiard: A Lyapunov exponents analysis

The dynamics of an ensemble of non-interacting particles suffering elastic collisions inside a driven stadium-like billiard is investigated through a four-dimensional nonlinear mapping. The system presents a resonance velocity, which plays an important role on the ensemble separation according to the initial velocities. The idea is to use the Lyapunov exponents to study the ensemble separation. We compare two different methods: (i) Finite Time Lyapunov Exponents and (ii) Wolf Linearization Algorithm. Results from both methods show good agreement regarding the separation of ensembles. Our results give support for the fact that stickiness influences the dynamics providing a crucial effect in the ensemble separation of the velocities for the time-dependent stadium-like billiard.

Thermodynamics of a time-dependent and dissipative oval billiard: A heat transfer and billiard approach

We study some statistical properties for the behavior of the average squared velocity-hence the temperature-for an ensemble of classical particles moving in a billiard whose boundary is time dependent. We assume the collisions of the particles with the boundary of the billiard are inelastic, leading the average squared velocity to reach a steady-state dynamics for large enough time. The description of the stationary state is made by using two different approaches: (i) heat transfer motivated by the Fourier law and (ii) billiard dynamics using either numerical simulations and theoretical description.

Acceleration in a nonplanar time-dependent billiard.

We study the dynamical properties of a particle in a nonplanar square billiard. The plane of the billiard has a sinusoidal shape. We consider both the static and time-dependent plane. We study the affect of different parameters that control the geometry of the billiard in this model. We consider variations of different parameters of the model and describe how the particle trajectory is affected by these parameters. We also investigate the dynamical behavior of the system in the static condition using its reduced phase plot and show that the dynamics of the particle inside the billiard may be regular, mixed, or chaotic. Finally, the problem of the particle energy growth is studied in the billiard with the time-dependent plane. We show that when in the static case, the billiard is chaotic, then the particle energy in the time-dependent billiard grows for a small number of collisions, and then it starts to saturate. But when the dynamics of the static case is regular, then the particle average energy in the time-dependent situation stays constant.

Evolution to the equilibrium in a dissipative and time dependent billiard

Abstract We study the convergence towards the equilibrium for a dissipative and stochastic time-dependent oval billiard. The dynamics of the system is described by using a generic four dimensional nonlinear map for the variables: the angular position of the particle, the angle formed by the trajectory of the particle with the tangent line at the position of the collision, the absolute velocity of the particle, and the instant of the hit with the boundary. The dynamics of the stationary state as well as the dynamical evolution towards the equilibrium is made by using an ensemble of non interacting particles. Finally, we make a connection with the thermodynamic by using the energy equipartition theorem.

Universal energy diffusion in a quivering billiard.

We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.

Time-dependent quantum circular billiard

The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schrodinger equation with time-dependent boundary conditions. The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disap- pears are explored in detail. The exact analytical solutions for monotonically expanding and contracting circles are obtained. For all cases, the time-dependence of the quantum average energy is calculated. It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation. For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken. However, for very high frequencies, the average energy once again becomes periodic. A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.

Exponential Fermi acceleration in general time-dependent billiards.

We show, that under very general conditions, a generic time-dependent billiard, for which a phase space of corresponding static (frozen) billiards is of the mixed type, exhibits the exponential Fermi acceleration in the adiabatic limit. The velocity dynamics in the adiabatic regime is represented as an integral over a path through the abstract space of invariant components of corresponding static billiards, where the paths are generated probabilistically in terms of transition-probability matrices. We study the statistical properties of possible paths and deduce the conditions for the exponential Fermi acceleration. The exponential Fermi acceleration and theoretical concepts presented in the paper are demonstrated numerically in four different time-dependent billiards.

Separation of particles leading either to decay or unlimited growth of energy in a driven stadium-like billiard

Competition between the decay and growth of energy in a time-dependent stadium billiard is discussed with emphasis on the decay of the energy mechanism. A critical resonance velocity is identified as causing the separation between ensembles of high and low energy and a statistical investigation is performed using ensembles of initial conditions both above and below the resonance velocity. For high initial velocity, Fermi acceleration is inherent in the system. However, for low initial velocity, the resonance and stickiness hold the particles in a regular or quasi-regular regime near the fixed points, preventing them from exhibiting Fermi acceleration. Also, a transport analysis along the velocity axis is discussed to quantify the competition in the growth and decay of the energy, making use of the distributions of histograms of frequency, and we find that the energy decay is caused by the capture of the orbits by the resonant fixed points.

Fermi acceleration in chaotic shape-preserving billiards.

We study theoretically and numerically the velocity dynamics of fully chaotic time-dependent shape-preserving billiards. The average velocity of an ensemble of initial conditions generally asymptotically follows the power law 〈v〉 = n(β) with respect to the number of collisions n. If a shape of a fully chaotic time-dependent billiard is not preserved it is well known that the acceleration exponent is β = 1/2. We show, on the other hand, that if a shape of a fully chaotic time-dependent billiard is preserved then there are only three possible values of β depending solely on the rotational properties of the billiard. In a special case in which the only transformation is a uniform rotation there is no acceleration, β = 0. Excluding this special case, we show that if a time-dependent transformation of a billiard is such that the angular momentum of the billiard is preserved then β = 1/6, while β = 1/4 otherwise. Our theory is centered around the detailed study of the energy fluctuations in the adiabatic limit. We show that three quantities, two scalars and one tensor, completely determine the energy fluctuations of the billiard for arbitrary time-dependent shape-preserving transformations. Finally, we provide several interesting numerical examples, all in a perfect agreement with the theory.

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.

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.

A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework, the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions n is determined by the Fokker-Planck equation (FPE). In the literature, the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions have been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a standard simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.

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.

Dynamical properties of the Fermi-Ulam model with external perturbation of Van der Pool type

In this work we consider the dynamics of a classical particle of mass m confined within an one-dimensional time dependent billiard. We assume that the time perturbation of the boundary is given by a nonlinear Van der Pool oscillator. The particle is confined to move between the two rigid walls, one of them fixed at a distance L from the first that fluctuates according to the Van der Pol oscillator. We construct the phase space and explore the dynamic properties of the system. The Van der Pol oscillator is a modification of the harmonic oscillator where a dissipative and nonlinear term is introduced. The equation describing the oscillator is M d^2x/dt^2 + b(x^2 − x0^2 )dx/dt + kx = 0 where b is a dissipative term, M is the mass, and k is the Hook constant.

Resonant population transfer in the time-dependent quantum elliptical billiard

We analyze the quantum dynamics of the time-dependent elliptical billiard using the example of a certain breathing mode. A numerical method for the time propagation of an arbitrary initial state is developed based on a series of transformations, thereby removing the time dependence of the boundary conditions. The time evolution of the energies of different initial states is studied. The maximal and minimal energies that are reached during the time evolution show a series of resonances as a function of the applied driving frequency. At these resonances, higher (or lower) lying states are periodically populated, leading to the observed change in energy. The resonances occur when the driving frequency or a multiple of it matches the mean energetic difference between the two involved states exactly. This picture is confirmed by a few-level Rabi-like model with periodic couplings, reproducing the key results of our numerical study.

Energy gain induced by boundary crisis

We consider a nonlinear system and show the unexpected and surprising result that, even for high dissipation, the mean energy of a particle can attain higher values than when there is no dissipation in the system. We reconsider the time-dependent annular billiard in the presence of inelastic collisions with the boundaries. For some magnitudes of dissipation, we observe the phenomenon of boundary crisis, which drives the particles to an asymptotic attractive fixed point located at a value of energy that is higher than the mean energy of the nondissipative case and so much higher than the mean energy just before the crisis. We should emphasize that the unexpected results presented here reveal the importance of a nonlinear dynamics analysis to explain the paradoxical strategy of introducing dissipation in the system in order to gain energy.

Velocity-dependent KAM islands

We revisit the non-dissipative time-dependent annular billiard and we consider the chaotic dynamics in two planes of conjugate variables in order to describe the behavior of the growth, or saturation, of the mean velocity of an ensemble of particles. We observed that the changes in the 4-d phase space occur without changing any parameter. They occur depending on where the initial conditions start. The emerging KAM islands interfere in the behavior of the particle dynamics especially in the Fermi acceleration mechanism. We show that Fermi acceleration can be suppressed, without dissipation, even considering the non-dissipative energy context.

Particle focusing in oscillating dissipative billiards

We develop and analyze a scheme to achieve both spatial and energetic focusing of an ensemble of neutral particles which is based on an oscillating billiard with frictional forces. The interplay of two competing mechanisms, acceleration due to collisions with the oscillating billiard walls and deceleration caused by friction, leads to the emergence of attractors in phase space. Their specific properties, i.e., spatial localization and energy spread, can be controlled and tuned by varying, e.g., the frequency of the time-dependent billiard.