Lennard-Jones流體之瞬間正則模頻譜內遷移邊界的多重碎形分析

Abstract

在本論文中，探討在Lennard-Jones位能交互作用下的流體內瞬間正則模之局域至非局域轉變，即系統由局域模態到擴張模態的轉變。而在瞬間正則模頻譜中，分別存在有實頻與虛頻瞬間正則模之局域至非局域遷移邊界。我們利用瞬間正則模在遷移邊界上表現出多重碎形的特性，即奇異頻譜及振動幅度的機率密度函數在遷移邊界上不隨系統大小改變， 就此特性進行推估遷移邊界所在位置的頻率。但實頻遷移邊界在計算上仍然有因系統效應難以克服之問題，無法準確計算出遷移邊界，因此本文主要探討虛頻瞬間正則模遷移邊界。我們的結果證實奇異頻譜幾乎與安德森模型在臨界亂度及在一簡單短程斥力流體內正則模之遷移邊界具有良好的一致性。在不同的熱力學狀態下，正則模遷移邊界也仍然具有一致相同的多重碎形特性，且在振動幅度的機率密度函數上的最大值具有不隨系統大小改變的特性。

In this thesis, we have investigated the localization-delocalization transitions (LDTs) of the instantaneous normal modes (INMs) in simple fluids with Lennard-Jones (LJ) potential. The LDT is the transition of the instantaneous vibrations in the system from the localized to extended states. The INM spectrum of a simple fluid consists of the real- and imaginary-frequency branches, corresponding to the positive and negative eigenvalues. The multifractal properties of the INMs at a mobility edge (ME) show that the singularity spectrum (SSP) and the probability density function (PDF) of vibrational amplitudes are invariant with the system size. Therefore, we use these properties to locate the ME in an INM frequency spectrum. Since the multifractal analysis for the ME in the real-frequency branch still has the formidable system size effect, we are not able to locate the ME precisely. So, we only consider the ME in the imaginary-frequency branch. Our results indicate that the singularity spectrum of the multifractal INMs almost agrees with that of the Anderson Model (AM) at the critical disorder and that of the INMs at the ME of the short-range truncated Lennard-Jones fluid. Also, for the LJ fluids at the thermodynamic states that we have simulated, the SSP of the multifractal INMs still has the same agreement. This agreement is a numerical evidence for the universality of the multifractals at the LDT. Besides this, within numerical errors, the location of the maximum in the PDF of vibrational amplitudes is also evidenced to be invariant with the system size.