We develop a method to calculate the configuration-averaged density of phonon modes in a liquid. Our strategy is based on the isomorphism between the calculation of the density of modes of a harmonic structure and the determination of transport properties of a random walker on that structure. The density of modes calculation for a fluid in d dimensions is shown to require solution of a random walk, in which a walker with d internal states moves among sites located at the particles of the fluid. We generalize the random walk theory of Gochanour, Andersen, and Fayer to treat this vector random walk, and use this approach to calculate the averaged density of phonon modes in a Lennard-Jones fluid. The calculation agrees well with Monte Carlo simulation results of Seeley and Keyes.
We have investigated the effects of local structures on the orientational motions in liquid water in terms of the instantaneous normal mode (INM) analysis. The local structures of a molecule in liquid water are characterized by two different kinds of index: the asphericity parameter of its Voronoi polyhedron and the numbers of the H bonds donated and accepted by the molecule. According to the two kinds of index, the molecules in the simulated water are classified into subensembles, for which the rotational contributions to the INM spectrum are calculated. Our results indicate that by increasing the asphericity, the rotational contribution has a shift toward the high-frequency end in the real spectrum and a decrease in the fraction of the imaginary modes. Furthermore, we find that this shift essentially relies on the number of the donated H bonds of a molecule, but has almost nothing to do with that of the accepted H bonds. The local structural effects resulting from the geometry of water molecule are also discussed.
In terms of the multifractal analysis, we investigate the characteristics of the instantaneous normal modes (INMs) at two mobility edges (MEs) of a simple fluid, where the locations of the MEs in the INM spectrum were identified in a previous work [B. J. Huang and T. M. Wu, Phys. Rev. E 79, 041105 (2009)]. The mass exponents and the singularity spectrum of the INMs are obtained by the box-size and system-size scalings under the typical average. The INM eigenvectors at a ME exhibit a multifractal nature and the multifractal INMs at each ME yield the same results in generalized fractal dimensions and singularity spectrum. Our results indicate that the singularity spectrum of the multifractal INMs agrees well with that of the Anderson model at the critical disorder. This good agreement provides numerical evidence for the universal multifractality at the localization-delocalization transition. For the multifractal INMs, the probability density function and the spatial correlation function of the squared vibrational amplitudes are also calculated. The relation between the probability density function and the singularity spectrum is examined numerically, so are the relations between the critical exponents of the spatial correlation function and the mass exponents of the multifractal INMs.
Thoracic PVCR can lead to satisfactory outcomes in the treatment of severe spinal deformities. Risk factors for neurological complications include the age over 18 years, presence of pulmonary dysfunction, and EBL greater than 50%. The pulmonary dysfunction can be regarded as the most valuable indicator to measure the severity of the spine deformity.
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