Chaos And Fractal Analysis Of Two-dimensional Mesoscopic Quantum Billiards Based On The RIKEN Devices

Chaos theory is a new type of non-linear theory developed in recent years, and fractal is anuseful tool to analysis the graph in chaotic phenomena. Both of them have important significancein many areas. The chaos and fractal are widely contacted with meteorology, geology and manysubdisciplines in physics, and play an important role in solving the non-linear problems.Different from the deterministic by Newton’s classical mechanics, chaotic system has a highdegree of dependence on initial conditions. However, by the limit of the measurement tool andthe initial conditions can not measured accurately, the chaotic trajectories can not be determinedaccurately, especially in the process away from the initial moment. Unlimited fine structureexists in chaotic system, there is always a new structure appeared in a new scale range, but thechaos unchanged. To study the regularity in chaotic random acts, we generally use the fractaltheory. Self-similar structures exist in scaleless range in chaotic curve, wherein the partial area isconsistent with the original curve after it was enlarged accordance with the similar ratio.Self-similar structures plays an important part in fractal theory, has great significance to thefractal of the chaotic system. There are different fractal research methods for different chaoticsystems. Unlike the exact self-similar structures mathematical constructed, the fractal theory isbuild on the basis of statistical physics.Chaos and fractal in two-dimensional mesoscopic billiards cavity is based on the particlestransport processes in this paper. With the development of lithography and the growth of crystalin recent twenty years, it is possble to make a micron scale two-dimensional mesoscopic billiards.Many phenomenons and behaviors exist chaos and fractal while particles’ transport processes inthe system of two-dimensional mesoscopic devices, by changing devices’ parameters orenvironment (such as temperature, magnetic field, etc.) to explore the influence to the escapecurves’ chaotic nature, and we also can found self-similar fractal structures in them. Thetheoretical model of RIKEN mesoscopic devices in our study is one kind of the two-dimensional Sinai billiards, which is an ideal model to investigate the chaotic and fractal behaviors in particles’ escapecurves because of the generation principle of chaos is relatively simple, and it is also the maindevice in this paper.This paper is divided into five chapters. The first chapter describes the chaos, fractal and itsrelated concepts’ production and development. The second chapter describes thetwo-dimensional mesoscopic billiards system, especially the RIKEN devices including it’stheoretical model and parameters; proposed the sensitive dependence on initial conditions whenthe particles escape from the RIKEN devices cavity; introduced the research progress of therelated fields at home and abroad. In the third chapter, we researched the chaotic nature of theescape curve by two methods which are the escape curves’ qualitative comparison and the fractaldimensions’ quantitative calculation, and got the impact to the escape curves’ chaotic nature bythe device parameters such as opening width, cavity length, corner positions, arc radius, etc. Inchapter four, we found the scaleless range in the escape curves and there are many goodself-similar structures in it; we firstly researched the relationship between self-similar structuresand its scale by the “eye-style structure” analysis; proved the constancy of the similar ratios in asame scaleless range, then we got a new method for the quantitative characterization of chaos.The fifth chapter is the summary and prospects to the works during graduate student.