以RLC共振電路類比研究二維晶格振盪 與量子混沌之特性

Abstract

在本篇論文中，從自然界共振的現象進入，介紹出一些與共振相關的具體模型，方形的侷限邊界下，離散系統如耦合振子與RLC電路，連續的系統下以本徵函數展開的Green函數。晶格的振盪行為由耦合振子來認識，RLC電路可以用來研究共振模態。建立在方形邊界下之二維網絡，推導出耦合振子之共振模態的數學形式，能與RLC共振電路類比，在模態的形貌與頻率上有好的對應。藉著數學軟體視覺化的呈現，讓我們能清楚RLC共振電路中的物理。阻尼的效應帶來的影響，並離散系統的色散關係，文中做了詳盡的討論。量子渾沌的領域中討論了許多不可積系統之邊界，藉著RLC電路可以方便的模擬出任意形狀的邊界。波函數與能階的統計行為，更呈現出系統的渾沌的特性。而連續體中的共振模態，以非齊次的Helmholtz方程式來描述，Green函數成為這類問題很好的工具。最後，離散與連續的系統作了一些比較，並發現若要以RLC電路去類比研究量子渾沌的現象，在低階的模態會相當接近。這份研究工作使我們看見這幾個不同模型間的類比關係，與在量子渾沌的研究中的類比情形。RLC共振電路的研究相當有趣，且仍有許多發展空間。

In this thesis, we start our research works with an introduction of phenomena of resonance in nature. There are many models in the world, we take some classical examples including coupled oscillators and RLC circuits to show wave phenomena in a discrete system. Then, Green function is introduced to explore continuous wave for resonant systems. From a historic view, we can see much interests on these topics. Inspired by previous works, we follow the patterns and set up a two-dimensional network for both coupled oscillators and RLC circuits. With a lot of efforts to develop the model of network of oscillators, we can easily obtain some visualized results through math software. Comparing the model with RLC electric network, we find a wonderful analog between these two models. In addition, we investigate influences of damping effect on the mode patterns of these two models and also discuss the dispersion relation in the discrete system in detail. RLC circuits can generate spatial patterns of arbitrary shapes, whose statistical properties can be analyzed quantitatively in the field of quantum chaos. There are many classical examples in quantum chaos. In theoretical analysis, we seek solution of inhomogeneous Helmholtz equation by Green function. And we find that there is a valid analog between Green function and RLC circuits in an approximately proportion of range of resonant frequencies of RLC circuits. The thesis show us a picture of these different models with comparisons and analysis. This work is interesting that there will be more knowledge of study of RLC circuits or more discovery in this filed in the future.