В рамках модели устойчивых случайных матриц, обладающих трансляционной инвариантностью, рас-смотрена двумерная (на квадратной решетке), неупорядоченная колебательная система со случайными, сильно флуктуирующими связями. Путем численного анализа динамического структурного фактора S(q, ω) показано, что колебания с частотами ниже частоты Иоффе−Регеля ωIR представляют собой обычные фононы с линейным законом дисперсии ω(q) ∝ q и обратным временем жизни Ŵ ∼ q 3 . Колебания же с частотами выше частоты ωIR хотя и являются делокализованными, не могут быть описаны плоскими волнами с определенным законом дисперсии ω(q). Они характеризуются диффузионным структурным фактором с обратным временем жизни Ŵ ∼ q 2 характерным для диффузионного процесса. В литературе их часто называют диффузонами. Показано, что как и в трехмерной модели, бозонный пик на частоте ωb в приведенной плотности колебательных состояний g(ω)/ω порядка частоты ωIR. Он расположен в переходной области между фононами и диффузонами и пропорционален модулю Юнга решетки ωb ≃ E. Я.М. Бельтюков благодарит Совет по грантам Президента РФ за финансовую поддержку (стипендия СП-3299.2016.1).
The random matrix theory is applied to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown that the most significant mechanical properties of amorphous solids can be taken into account using the correlated Wishart ensemble. In this ensemble, an excess vibrational density of states over the Debye law is observed as a peak in the reduced density of states g(ω)/ω. Such a peak is known as the boson peak, which was observed in many experiments and numerical simulations for two-dimensional and three-dimensional disordered systems. It is shown that two-dimensional systems have a number of differences in the asymptotic behavior of the boson peak.
In the random matrix model with translational symmetry, it is studied the influence of nanoparticles on the macroscopic rigidity of an amorphous system. The numerical analysis shows that the macroscopic theory of elasticity can be applied if the radius of nanoinclusions R is big enough. In this case, it gives an additional contribution to the Young's modulus ΔE ~ R³. However, as the radius of nanoinclusions decreases, this dependence becomes quadratic, ΔE ~ R². Theoretical result for Young's modulus was obtained by reducing the energy of the whole system to a sum of quadratic forms and by applying the Gauss–Markov theorem. From this theorem, it follows that the stiffness of the system depends on the difference between the number of bonds and the number of degrees of freedom, which is proportional to the surface area of nanoinclusions. It is shown, that there is a scale of nanoinclusion radius, which characterizes a scale of inhomogeneity of amorphous solids. It determines the smallest characteristic size of nanoinclusions, at which the macroscopic theory of elasticity can be applied.
The statistical features of quasi-local vibrations of disordered systems are studied within the framework of the model of correlated random matrices. It is shown that the statistics of matrix elements of the dynamical matrix strongly affects the properties of such vibrations. The lowest frequency part of the density of states of quasilocal vibrations is described by the expression ρ_qlv(ω) ∝ ω^n, where the power n is the number of neighboring atoms. However, if the distribution of matrix elements is highly non-Gaussian, an additional dependence ρ_qlv(ω) ∝ ω^γ appears, where the power γ decreases as the degree of non-Gaussianity increases.
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