Norio Inui

Squared deviations from the equilibrium positions of one-dimensional coupled harmonic oscillators with fixed and free endpoints are calculated, and the time averages are expressed as a function of the initial displacements and velocities. Furthermore, we consider the averages of squared deviations over an ensemble of initial displacements and velocities, which distribute based on a product of the same distribution functions with variances of the initial coordinates σ x 2 and velocities σ v 2 , respectively. We demonstrate that the mean squared deviation linearly increases as the oscillator separates further from the fixed endpoint because of the asymmetrical boundary conditions, and that the increase rate depends only on σ v 2 and not on σ x 2 . This simple statistical property of harmonic oscillation is similarly observed in the oscillations of a graphene sheet and carbon nanotube in molecular dynamics simulations, in which the interacting forces are nonlinear.

Graphene exhibits diamagnetism, and its origin is the orbital electric currents induced on the surface by an applied magnetic field. The magnetic response of a graphene cantilever in the presence of a magnetic field is mainly determined by the diamagnetic electric current, and spin paramagnetism, which suppresses the diamagnetism. We elucidate the change in the electric current distribution caused by the large bending of the graphene cantilever using the tight-binding model. The electric current almost disappears when the position of the graphene cantilever transitions from perpendicular to parallel to the magnetic field and reverses when the graphene cantilever is folded in half. Furthermore, the temporal change in the magnetic energy of the vibrating graphene cantilever is calculated using the molecular dynamics simulation. The strong dependence of the magnetization of a graphene cantilever on its position relative to the magnetic field can be utilized for actuating and controlling the cantilever.

This study considers the energy level of a charged particle on a large hexagonal lattice in a magnetic field. The discretized Schrödinger equation on a hexagonal lattice, which can be expressed as a special case of a tight-binding model is derived, and its energy level is numerically calculated. The size dependence of the energy level near zero for large radii is considered by analyzing the asymptotic behavior of the zeros of the Laguerre function, which is the radical wavefunction of the continuous Schrödinger equation. Additionally, the splitting of the Landau level due to the finite size of a hexagonal disk is discussed.

Graphene exhibits diamagnetism, enabling it to be lifted by the repulsive force produced in an inhomogeneous magnetic field. However, the stable levitation of a graphene flake perpendicular to the magnetic field is impeded by its strong anisotropic of magnetic susceptibility that induces rotation. A method to suppress this rotation by applying the Casimir force to the graphene flake is presented in this paper. As a result, the graphene flake can archive stable levitation on a silicon plate when the gravitational force is small.