Universality in the vibrational spectra of weakly-disordered two-dimensional clusters

Abstract

We numerically investigate the vibrational spectra of single-component clusters in two dimensions. Stable configurations of clusters at local energy minima are obtained, and for each the Hessian matrix is evaluated and diagonalized to obtain eigenvalues as well as eigenvectors. We study the density of states so obtained as a function of the width of the potential well describing the two-body interaction. As the width is reduced, as in three dimensions, we find that the density of states approaches a common form but the two-peak behavior survives. Further, calculations of the participation ratio show that most states are extended, although a smaller fraction of the degrees of freedom are involved in these modes compared to three dimensions. We show that the fluctuation properties of these modes converge to those of the Gaussian orthogonal ensemble of random matrices, in common with previous results on three-dimensional amorphous clusters and molecular liquids.

Spontaneous back-pain alters randomness in functional connections in large scale brain networks: A random matrix perspective

“…This ensures all the eigenvalues are on same footing. The analytical fitting function used for unfolding need not be unique and, is generally different for different systems [30][31][32][33][34]. For this study, the eigenvalue spectra of all the correlation matrices generated is approximated extremely well by a function of the form…”

“…Figure 2 shows a plot of the cumulative eigenvalue density along with the analytical fitting function. We leave out a small portion of eigenvalues at the upper end (3 or 4 eigenvalues) in order to achieve the best fit, something which has been a standard practice in other works [30][31][32][33][34]. We deal with unfolded eigenvalues from this point onwards.…”

“…In such a scenario, one settles for a model that captures the statistical properties of the energy spectrum. RMT finds extensive applications in the statistical studies of various complex systems such as quantum chaotic systems, complex nuclei, atoms, molecules, disordered mesoscopic systems [16][17][18][19][20][21][22][23][24], atmosphere [25], financial applications [26], complex networks [27], societal networks [28], network forming systems [29,30], amorphous clusters [31][32][33][34], biological networks [35], protein networks [36,37], and cancer networks [38] etc. In recent years, RMT has also been applied towards brain network studies in studying universal behavior of brain functional connectivity and has been effective in detecting the differences in resting state and visual stimulation state [39,40].…”

Spectral statistics of the quenched normal modes of a network-forming molecular liquid

“…[13][14][15][16]26,27 To unfold the data, we define H͑͒ to be the number of frequencies equal to or less than ͑shown in Fig. Unless stated otherwise, the fluctuation properties presented below are computed for the eigenfrequencies, in contrast to earlier studies 13-16 that used the eigenvalues.…”

“…[11][12][13][14][15][16] These results have been obtained using the spectra derived from both INM and QNM properties, although the clearest indications seem to have come from using QNM data. [11][12][13][14][15][16] These results have been obtained using the spectra derived from both INM and QNM properties, although the clearest indications seem to have come from using QNM data.…”