Abstract: | The random matrix theory has been used for analyzing vibrational spectra of amorphous solids. The random dynamical matrix
M = AA
T
with nonnegative eigenvalues ɛ = ω2 has been investigated. The matrix A is an arbitrary square (N-by-N) real sparse random matrix with n nonzero elements in each row, mean values 〈A
ij
〉 = 0, and finite variance 〈A
ij
2〉 = V
2. It has been demonstrated that the density of vibrational states g(ω) of this matrix at N, n ≫ 1 is described by the Wigner quarter-circle law with the radius independent of N. For n ≪ N, this representation of the dynamical matrix M = AA
T
makes it possible in a number of cases to adequately describe the interaction of atoms in amorphous solids. The statistics
of levels (eigenfrequencies) of the matrix M is adequately described by the Wigner surmise formula and indicates the repulsion of vibrational terms. The participation
ratio of the vibrational modes is approximately equal to 0.2–0.3 almost over the entire range of frequencies. The conclusions
are in qualitative and, frequently, quantitative agreement with the results of numerical calculations performed by molecular
dynamics methods for real amorphous systems. |