在一簡單流體內瞬間正則模之局域至非局域轉變

Abstract

在本文中，我們探討在一簡單短距交互作用流體內瞬間正則模之局域至非局域轉變。此液體模型提供拓樸性無序系統的原形。在瞬間正則模頻譜中，分別在正數與負數瞬間正則模頻譜上發現轉變點。我們使用有限尺度標度法定出轉變點，並且計算關聯長度的臨界指數。在數值誤差範圍內，所估計的臨界指數與安德森模型是一致的。此結果驗證了三維的安德森模型與拓樸性無序系統的短距簡單流體屬於同樣的統計普適性分類。 我們也對瞬間正則模做了多重碎形的分析。在局域與非局域轉變的轉變點附近，瞬間正則模表現出多重碎形的特性。用廣義碎形維度與波分量強度頻譜可以觀測到在轉變點上隨著系統尺度的不變性。我們精確計算了波分量強度頻譜，在轉變點上，我們的結果與安德森模型有高度的一致性，證明了波分量強度頻譜也是一個具有普適性的量。

In this thesis, we have investigated the localization-delocalization transitions (LDTs) of the instantaneous normal modes (INMs) in a simple□fluid with short-ranged interactions. The model□fluid is a prototype of topologically disordered systems. Two LDTs in the INM spectrum are found, and the locations are termed as the mobility edges (MEs) with one in the positive-eigenvalue branch and the other in the negative-eigenvalue one. The MEs and the critical exponents of the two LDTs are estimated by the…finite-size scaling (FSS) for the second moments of the nearest-neighbor level-spacing (LS) distributions. Within numerical errors, the two estimated critical exponents are almost coincident with each other and close to that of the Anderson model (AM) in three dimensions. The nearest-neighbor LS distribution at each ME is examined to be in a good agreement with that of the AM at the critical disorder. We conclud that the LDTs in the Hessian matrices of topologically disordered systems exhibit the critical behaviors of orthogonal universality class. We also investigate the analysis of level-number variance (LNV) and level compressibility (LCP), which characterizes the nature of the correlation of energy levels beyond the mean LS. Furthermore, in terms of the multifractal analysis, the INM eigenvectors exhibit a multifractal nature with the same generalized fractal dimensions and the sigularity spectrum. Our results indicate that the singularity spectrum of the multifractal INMs agrees with that of the AM at the critical disorder. This good agreement provides a numerical evidence for the universality of the multifractals at the localization-delocalization transition. With the multifractal INMs, we calculate the probability den- sity funciton and the spatial correlation function of the squared vibrational amplitudes. With the multifractal INMs, the relation between the probability density function and the singularity spectrum is examined, so are the relations between the critical exponents of the spatial correlation function and the generalized fractal dimensions.