Phonons and the Ioffe–Regel crossover in random lattices with arbitrary dimension

In the frame of the random matrix theory, it was shown, that the relaxation of the projection of the initial plane wave with the wave vector $\mathbf{q}$ is described by the equation of motion with the memory function which corresponds to the complex dynamical Young modulus $E(\omega)$. In the harmonic scalar model of displacements with the absence of energy dissipation, the Ioffe–Regel crossover arises universally in amorphous systems with the dimension $d\ge3$. Vibrations above the Ioffe–Regel crossover are related to the diffusive nature and can be described by the diffusion equation with the damping $\Gamma(\mathbf{q})\propto q^2$.