振動在局限與非局限轉變之多重碎形分析(II)

Abstract

安德森侷限是波物理中普遍的現象，物理意義是波被在所傳遞介質內的無序所侷限。安德森侷限是一有趣的題目，目前仍有許多相關的研究在進行。最近，在由鋁顆粒焊接起來的三維空間的彈性連結網，超音波的侷限已被實驗觀察到。這激起我們探討振動波被侷限在非晶格的無序系統內。 我們最近發表了一篇論文，是有關在TLJ流體瞬間正則模頻譜內的侷限與非侷限轉變。TLJ流體内粒子間的作用位能只是LJ位能的斥力部分。藉由態能階間距的統計及有限系統的縮尺理論，我們找到兩個侷限與非侷限轉變在瞬間正則模頻譜內，一個為正的固有質，另一個為負的固有質。在數值誤差範圍內，這兩個侷限與非侷限轉變的臨界指數及最鄰近態間距分佈函數相一致，且與安德森模型在三維空間的結果接近。這符合侷限與非侷限轉變的普遍性，和隨機矩陣理論的預期一致。 在本計劃內，我們改用多重碎形分析方法，近一步探討瞬間正則模在侷限與非侷限轉變附近的性質。基本上，瞬間正則模在侷限與非侷限轉變具有多重碎形的性質，其特性可用奇異性頻譜表現。利用標準的箱子計數法，我們將計算瞬間正則模在這兩個侷限與非侷限轉變的奇異性頻譜，而所得結果將與安德森模型結果比對。我們也檢驗流體模擬系統的大小對此奇異性頻譜的影響。當粒子間的作用位能恢復為LJ位能時，可檢驗吸力部分對兩個侷限與非侷限轉變位置的改變。最後，奇異性頻譜與瞬間正則模振幅平方分佈函數之間的關係也一併討論。

Anderson localization (AL) is a general phenomenon associated with the localization of waves due to the disorder in the medium that the waves propagate through. After fifty years, AL is still an interesting topic with active researches. Recently, the localization of ultrasound is observed in a three-dimensional elastic network composed of Al beads brazed together. This intrigues us to investigate the localization of vibrations in topologically disordered systems. We have recently published a paper with regard to the localization-delocalization transition (LDT) in the instantaneous-normal-mode (INM) spectrum of a fluid with the truncated Lennard-Jones (TLJ) potential, which is only the repulsive part of the LJ potential. By the level-spacing (LS) statistics and the approach of finite-size scaling, we find two LDTs in the INM spectrum, with one in the positive-eigenvalue branch and the other in the negative-eigenvalue branch. Within numerical errors, the critical exponents and the functions of the nearest-neighbor distribution at the two LDTs agree with each other and are close to those of the Anderson model (AM) in three dimensions. These results are consistent with the prediction of the random matrix theory for the universality at the LDT. In this project, we use the multifrartal analysis to further investigate the properties of the INMs near the two LDTs. Generally, the INMs at the LDTs exhibit a multifractal nature, which is characterized by the singularity spectrum. With the standard box-counting method, we will calculate the singularity spectra of the INMs at the two LDTs and also examine the size effects on the singularity spectra. The calculated results will be compared with that of the AM. We will study how the locations of the two MEs are influenced by the attractions between the particles as the pair interactions are recovered back to the LJ potential. The relationship between the singularity spectrum and the probability density function of the squared vibrational amplitudes is also investigated.