On the statistical and transport properties of a non-dissipative Fermi-Ulam model

Abstract

The transport and diffusion properties for the velocity of a Fermi-Ulam model were characterized using the decay rate of the survival probability. The system consists of an ensemble of non interacting particles confined to move along and experience elastic collisions with two infinitely heavy walls. One is fixed, working as a returning mechanism of the colliding particles while the other one moves periodically in time. The diffusion equation is solved and the diffusion coefficient is numerically estimated by means of the averaged square velocity. Our results show remarkably good agreement of the theory and simulation for the chaotic sea below the first elliptic island in the phase space. From the decay rates of the survival probability, we obtained transport properties that can be extended to other nonlinear mappings, as well to billiard problems. We study the dynamics of an ensemble of non interacting particles moving constrained by two infinitely heavy walls, where one of then is moving periodically in time and the other is fixed. This problem, also known as Fermi-Ulam model, has application in many areas, including astrophysics, atom-optics, quantum mechanics, among others. The diffusive behaviour of the velocity, here set as the way the transport of orbits occurs in the phase space, is investigated considering transport properties obtained from the decay rate of the survival probability, defined by means of escape formalism. Since the system present mixed dynamics, stickiness phenomenon may influence the transport causing anomalous diffusion. In this study we developed an analytical approach for the diffusion coefficient along the transport through the chaotic sea considering escape rate formalism and survival probability analysis. The numerical results we obtained are in good agreement with the theory, and confirm the robustness of the formalism. The results obtained here can be extended to other similar dynamical systems.

Transition from normal to ballistic diffusion in a one-dimensional impact system

“…The most important aspect of this analysis is that the escape rate is very sensitive to the system dynamics [55,56]. For strongly chaotic systems the decay is typically exponential [57][58][59], while for systems that present a mixed phase space the decay can be slower, presenting a mix of exponential with a power law [9], or even stretched exponential decay [60].…”

Investigation of stickiness influence in the anomalous transport and diffusion for a non-dissipative Fermi–Ulam model

“…The most important aspect of this analysis is that the escape rate is very sensitive to the system dynamics. For strongly chaotic systems the decay is typically exponential [15][16][17][18] , while systems that present mixed phase space the decay can be slower, presenting a mix of exponential with a power law [19][20][21] , or even stretched exponential decay [22] . Indeed, when a non-exponential decay is observed the dynamics would require a long range correlation.…”

“…One wall is assumed to be fixed while the other one oscillates periodically in time. The phase space is mixed and contains periodic islands surrounded by a chaotic sea, which is limited by a set of invariant spanning curves [15,25] . This implies that we have a finite portion of the phase space for orbits to diffuse [15] , which prevents the dynamics to exhibit unlimited diffusion in the velocity.…”

“…The phase space is mixed and contains periodic islands surrounded by a chaotic sea, which is limited by a set of invariant spanning curves [15,25] . This implies that we have a finite portion of the phase space for orbits to diffuse [15] , which prevents the dynamics to exhibit unlimited diffusion in the velocity. The mechanics of the model leads to a complex variety of nonlinear phenomena in both conservative and dissipative dynamics [26][27][28][29][30] .…”

“…Since the accessible phase space have a finite portion for the orbit to diffuse, it can be divided into two regions of high and low energy regimes, that depends on the initial velocity of the ensemble. In previous studies [15,25] , only the lower ensemble of energy (basically composed by chaotic sea) was investigated, leaving aside the higher ensemble, which has more complicated dynamics with chains of islands, cantori and small portions of chaotic sea, and seems more interesting to be studied. So, in this paper we give focus to the higher ensemble, but not neglecting the lower one as well, yielding in a complete overview of the dynamical scenario for the FUM.…”