The Research On Fractal Nature Of Escape Particles In Mesoscopic Stadium Shaped Devices

The scale of mesoscopic system can be described as being between the size of a quantity ofatoms and of materials measuring micrometres. We have studied the nonlinear behavior of theparticles in non integrable system by stadium shaped device and it can be depicted bychaos and fractal. The extreme sensitivity in chaotic area to initial conditions is the mostprominent behavior in chaotic devices. There exist fractal structures for stadium shaped devices,which can be investigated to find the general rule of escape particles in chaotic systems.We have found the chaotic behaviors of the particles have strong connections with variousparameters in chaotic device through altering the chaotic characteristics by changing theparameters of stadium shaped devices and analyzing the effect of these variations of parameterson fractal dimensions which can quantitatively present the chaotic porperties. The escapeparticles in mesoscopic stadium shaped devices satisfy the exponential damping law. Moreover,the strength of the chaos, which can be presented by the fractal dimensions, is determined by theoutside shape. We have found a good agreement between the fractal dimensions and the escaperates in mesoscopic stadium shaped devices, and we revealed that the exponential law of escapeis subjected to the shape of devices. We count and fit the relationship between the escape ratesand the wave numbers of the particles. As is shown in the numerical results, the relation betweenthe escape rates and the wave numbers is a quadratic function, but the escape rates are notstrictly linear with the change of the energy. Furthermore, we analyzed the influence of thediffraction effect at the lead opening on the escape particles. Numerical results show that thediffraction effect make the escape rates increasing, and the evolution of the number of particlesno longer obeys the law of exponential decay in a short time, but submits to it again in a longtime.In order to further identify the general rule of behaviors of the particles in chaotic devices，we studied the escape particles in Hénon-Heiles system and found the self-similar structures which the escape time of particles change over the exit angles. We calculated the fractaldimensions and escape rate by changing the energy of the system and found the fractaldimensions were strictly linear with the escape rate. We varied the chaotic properties of thesystem by structuring a kind of Hénon-Heiles system and got similar results. This illustrates therelationship between the fractal dimensions and the chaotic escape rate is applicable to differentkinds of chaotic systems. The fractal dimensions can be served as a tool to study the features ofescape particles in chaotic system. In practical application, we can characterize the transportbehavior through studying the fractal dimensions of chaotic electronic devices.This paper is divided into five chapters. In the first chapter, we introduce the generalmethod of chaos and fractal. An important tool has been provided to study the behaviors ofparticles in the chaotic system-the fractal dimension. In the next chapter, we present themesoscopic theory and the development of experiments. Moreover, we elaborate the escapetheory of particles in chaotic system and the application value of this research. The third chapteris the key content of this paper, in which we give the description of the chaotic property and theescape rate in mesoscopic stadium shaped devices and find the linear relationship between them.In addition, we analyze the influence of the diffraction effect on the escape particles at the leadopening. In chapter four, the credibility of our results is further proved by applying our results inHénon-Heiles system and its transformation. The fifth chapter is the conclusions of our study andthe outlook on the future applications.