Kousuke Yakubo

Global phase diagram of one-dimensional graded diatomic elastic chains: a diagrammatic approach to identifying vibrational normal modes

We study vibrational normal modes in graded diatomic chains wherein the masses m1 of one type of atom vary linearly with the gradient c while those of the other type m2 remain constant, in order to examine the diatomic effect on one-dimensional graded elastic chains. By means of a band overlapping picture—a convenient diagrammatic approach—we found six distinct kinds of vibrational mode, four of which are localized and two extended. Depending on their characteristics, we are also able to categorize these modes into acoustic and optical modes. Furthermore, investigating transitions among these rich normal modes we construct the global phase diagrams of vibrational modes in the ω–c space. All results are verified by numerical calculations in finite size systems.

Fluctuations of the Order Parameter in Critical Complex Networks

The generalized Gumbel distribution function (GGDF) has been conjectured to describe fluctuating order parameters of a large class of finite critical systems. We study probability distribution functions (PDFs) of rescaled order parameters in finite complex networks near their critical points to clarify whether this conjecture holds for any critical system. Our numerical results show that the PDF for fitness-model networks near the critical point, which has the scale-free property, cannot be described by the GGDF with the real parameter a equal to π/2 while the PDF for non-scale-free Erdős–Renyi random graphs obeys it. We also discuss the origin of the discrepancy with the GGDF in the scale-free network.

Distribution of Fluctuating Fractal Dimensions of Finite Critical Systems

General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).

Anomalous Size Dependence of Inverse Participation Ratio of Eigenfunctions in Graded Elastic Lattices

We have studied the inverse participation ratio (IPR) denoted by P -1 of vibrational modes in one-dimensional graded elastic chains. The size dependence of IPR can be shown to derive from a quantum interpretation of vibrational modes. The quantum analogue of peculiar vibrational modes (gradons) to graded systems is established for the hump structure of the gradon front via the fact that the probability of a quantum particle is inversely proportional to its velocity. In this way, the envelope function can be determined analytically, and matches the mode pattern quite well. We find that the IPR exhibits an anomalous size ( N ) dependence and can be captured accurately by the relation: N P -1 = C 1 log N + C 2 , where C 1 and C 2 are constants. This interpretation is important in understanding a wide variety of properties of graded systems.

Variety of normal modes and their transition behaviors in graded elastic networks : Square networks with a diagonal gradient

We study vibrational excitations in graded elastic lattices modelled by coupled harmonic oscillators in a square lattice, in which the force constants or the vibrating masses vary along a diagonal ...

Localized vibrations in graded lattices: Gradons

We have studied localized vibrational modes in graded elastic lattices in which vibrating masses or nearest-coupling force constants vary linearly in a uniaxial direction. We found localized modes, called gradons, whose characteristics are peculiar to graded systems and are qualitatively different from localized states due to diffusive scattering in disordered media and confined excitations trapped by impurity potentials. The spectral properties and features of mode profiles of gradons are presented in a simple one-dimensional graded chain. In addition, it is shown that a variety of gradons can be excited in graded diatomic chains. We also demonstrate that a band overlapping picture is efficient to understand the variety of gradon modes and their transitions to extended phonon modes.

Energy Relaxation in Damped Two-Dimensional Graded Elastic Lattices

We have studied the relaxation rate of vibrational modes in damped two-dimensional graded mass lattices. The relaxation rate spectrum in the weak damping limit can be obtained analytically through a perturbation theory based on the semiclassical quantum analogue envelope function. We found dip or peak structures on the relaxation rate spectrum. The dip or peak structures can be described quantitatively by the asymptotic behavior of relaxation rate at the transition frequencies. The frequency dependence of the relaxation rate is qualitatively explained by the mode patterns of gradon modes. The validity of the analytic results is confirmed by numerical solution with weak damping. In the strong damping case, we need to retain higher-order perturbations. These results can be applied to the energy relaxation in analogous systems.

VIBRATIONAL EXCITATIONS IN GRADED ELASTIC CHAINS

We study vibrational excitations in linearly graded materials and/or systems. Graded systems demonstrate a unique spectrum and mode profiles, resulting in a new type of localization-delocalization transition. The nature of gradients can confine certain vibrational excitations, and redistribute them spatially. These features are contrasted to the two extreme cases of inhomogeneous system, i.e., periodically modulated system and randomly disordered system, We show in detail vibrational normal modes sustained by one dimensional graded force constant and graded mass networks, in particular, a unusual kind of modes called gradons. We propose an approach to study vibrational modes in a grades elastic system with the help of a series of homogeneous systems. Using this approach, we elaborate the features of the elastic gradons and the phonon-gradon transition.