Dissipação no modelo Fermi-Ulam - Sousa, Danila Fernandes Tavares de (2012)
In this work, we revisit the Fermi accelerator model, also known as Fermi-Ulam model. This model consists of a classical particle of unitary mass wich is confined to bounce between two rigid walls. One of them is fixed and the other one is assumed to move periodically in time. The particle collisions with the walls are assumed to be elastic. The description of the dynamic is made everytime the particle collides with the moving wall, so that we know the particle’s time and velocity at each collision needed to describe the dynamic of the system. Two versions for this model are studied: the complete and the simplified versions. In the simplified version, two walls of the model are assumed to be fixed. The Fermi-Ulam model is a conservative model because it preserves area of the phase space. Our analytical and numerical results for this conservative model are presented and discussed. Some dynamical properties for a particle suffering the action of a drag force are obtained for a dissipative Fermi-Ulam model. The dissipation is introduced via a viscous drag force, like a gas, wich is assumed to be proportional to any power of the velocity, F = −ηV γ . The dynamics of the models are described by two dimensional nonlinear mappings obtained via the solution of the second Newtons’ law of motion. We prove analytically that the decay of high energy is given by a continued fraction wich recovers the following expressions: (i) linear for γ = 1, (ii) exponential for γ = 2 and (iii) second degree polynomial type for γ = 1.5. For any value of γ, the numerical results shows a polynomial behavior for the velocity decay. Our results are discussed for both the complete and simplified versions. The phase spaces and the basin of attraction for some values of γ are obtained. Complementing our studies on this dissipative version of the Fermi-Ulam model, a mixed model was proposed. In this model, one particle travels through two different media. It started in a medium with no dissipation lets say vacuum and at some point it enters a region with a dissipative medium. The dissipation is also introduced by a viscous drag force, such that F = −ηV γ . In particular, for the study of the mixed model we use γ = 1 and γ = 2. The system is characterized by the ratio of the two medium length ξ. We show that there is a smooth transition of the velocity regime with ξ. We construct the phase spaces for the complete and simplified versions of the models. For the limiting cases, ξ = 0 or ξ = 1, the system behaves like one medium only.