Nonpersistence of resonant caustics in perturbed elliptic billiards

Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics —the ones whose tangent trajectories are closed polygons— are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

Billiards in a General Domain with Random Reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain D R(d) until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord ""picked at random"" in D, and we study the angle of intersection of the process with a (d - 1) -dimensional manifold contained in D.

The presence and lack of Fermi acceleration in nonintegrable billiards

The unlimited energy growth ( Fermi acceleration) of a classical particle moving in a billiard with a parameter-dependent boundary oscillating in time is numerically studied. The shape of the boundary is controlled by a parameter and the billiard can change from a focusing one to a billiard with dispersing pieces of the boundary. The complete and simplified versions of the model are considered in the investigation of the conjecture that Fermi acceleration will appear in the time-dependent case when the dynamics is chaotic for the static boundary. Although this conjecture holds for the simplified version, we have not found evidence of Fermi acceleration for the complete model with a breathing boundary. When the breathing symmetry is broken, Fermi acceleration appears in the complete model.

A family of stadium-like billiards with parabolic boundaries under scaling analysis

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Recurrence of particles in static and time varying oval billiards

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

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 proportional to-V; (ii) F proportional to-V-2 and (iii) F proportional to-V-delta with 1 < delta < 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 < delta < 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.

Separation of particles in time-dependent focusing billiards

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Dynamics of classical particles in oval or elliptic billiards with a dispersing mechanism

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Circular, elliptic and oval billiards in a gravitational field

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Spin accumulation encoded in electronic noise for mesoscopic billiards with finite tunneling rates

We study the effects of spin accumulation (inside reservoirs) on electronic transport with tunneling and reflections at the gates of a quantum dot. Within the stub model, the calculations focus on the current-current correlation function for the flux of electrons injected into the quantum dot. The linear response theory used allows us to obtain the noise power in the regime of thermal crossover as a function of parameters that reveal the spin polarization at the reservoirs. The calculation is performed employing diagrammatic integration within the universal groups (ensembles of Dyson) for a nonideal, nonequilibrium chaotic quantum dot. We show that changes in the spin distribution determine significant alterations in noise behavior at values of the tunneling rates close to zero, in the regime of strong reflection at the gates.

Second floor Billiards Room, Michigan Union

2 scans made - 1of2=scanner presets, 2of2=auto color correction per scanner