An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schr{\"o}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / {\Lambda} is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice {\Lambda}. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.Comment: 32 pages, LaTeX, minor corrections are mad

The Acceleration Mechanism of Resistive MHD Jets Launched from Accretion Disks

We analyzed the results of non-linear resistive magnetohydrodynamical (MHD) simulations of jet formation to study the acceleration mechanism of axisymmetric, resistive MHD jets. The initial state is a constant angular momentum, polytropic torus threaded by weak uniform vertical magnetic fields. The time evolution of the torus is simulated by applying the CIP-MOCCT scheme extended for resistive MHD equations. We carried out simulations up to 50 rotation period at the innermost radius of the disk created by accretion from the torus. The acceleration forces and the characteristics of resistive jets were studied by computing forces acting on Lagrangian test particles. Since the angle between the rotation axis of the disk and magnetic field lines is smaller in resistive models than in ideal MHD models, magnetocentrifugal acceleration is smaller. The effective potential along a magnetic field line has maximum around $z \sim 0.5r_0$ in resistive models, where $r_0$ is the radius where the density of the initial torus is maximum. Jets are launched after the disk material is lifted to this height by pressure gradient force. Even in this case, the main acceleration force around the slow magnetosonic point is the magnetocentrifugal force. The power of the resistive MHD jet is comparable to the mechanical energy liberated in the disk by mass accretion. Joule heating is not essential for the formation of jets.Comment: 15 pages, 15 figures, 1 table, accepted for publication in Ap

Pressure-induced phase transitions of halogen-bridged binuclear metal complexes R_4[Pt_2(P_2O_5H_2)_4X]nH_2O

Recent contrasting observations for halogen (X)-bridged binuclear platinum complexes R_4[Pt_2(P_2O_5H_2)_4X]nH_2O, that is, pressure-induced Peierls and reverse Peierls instabilities, are explained by finite-temperature Hartree-Fock calculations. It is demonstrated that increasing pressure transforms the initial charge-polarization state into a charge-density-wave state at high temperatures, whereas the charge-density-wave state oppositely declines with increasing pressure at low temperatures. We further predict that higher-pressure experiments should reveal successive phase transitions around room temperature.Comment: 5 pages, 4 figures embedded, to be published in Phys. Rev. B 64, September 1 (2001) Rapid Commu

Magnetic translation groups in an n-dimensional torus

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus.Comment: 29 pages, LaTeX2e, title changed, re-organized, to be published in Journal of Mathematical Physic

Resistive Magnetohydrodynamics of Jet Formation and Magnetically Driven Accretion

We carried out 2.5-dimensional resistive magnetohydrodynamic simulations to study the effects of magnetic diffusivity on magnetically driven mass accretion and jet formation. We found that (1) when the normalized magnetic diffusivity, is small, mass accretion and jet formation take place intermittently; (2) when diffusivity is middle, the system evolves toward a quasi-steady state; and the system evolves toward a quasi-steady state; and (3) when diffusivity is large, the accretion/mass outflow rate decreases with diffusivity and approaches 0. The results of these simulations indicate magnetic braking provide a mass accretion rate which is sufficient to explain the activity of AGNs.Comment: 24 pages, LaTex, 15 jpg figures include, accepted for PAS

Semi-Phenomenological Analysis of Dynamics of Nonlinear Excitations in One-Dimensional Electron-Phonon System

The structure of moving nonlinear excitations in one-dimensional electron-phonon systems is studied semi-phenomenologically by using an effective action in which the width of the nonlinear excitation is treated as a dynamical variable. The effective action can be derived from Su, Schrieffer and Heeger's model or its continuum version proposed by Takayama, Lin-Liu and Maki with an assumption that the nonlinear excitation moves uniformly without any deformation except the change of its width. The form of the action is essentially the same as that discussed by Bishop and coworkers in studying the dynamics of the soliton in polyacetylene, though some details are different. For the moving excitation with a velocity $v$, the width is determined by minimizing the effective action. A requirement that there must be a minimum in the action as a function of its width provides a maximum velocity. The velocity dependence of the width and energy can be determined. The motions of a soliton in p olyacetylene and an acoustic polaron in polydiacetylene are studied within this formulation. The obtained results are in good agreement with those of numerical simulations.Comment: 19 pages, LaTeX, 7 Postscript figures, to be published in J. Phys. Soc. Jpn. vol.65 (1996) No.

Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity

Saari's homographic conjecture in N-body problem under the Newton gravity is the following; configurational measure \mu=\sqrt{I}U, which is the product of square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the motion is homographic. Where m_k represents mass of body k and r_{ij} represents distance between bodies i and j. We prove this conjecture for planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use \zeta=3q_3/(2(q_2-q_1)) where q_k \in \mathbb{C} represents position of body k. Using r_1=r_{23}/r_{12} and r_2=r_{31}/r_{12} in intermediate step, we finally use \mu itself and \rho=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables \mu and \rho make our proof simple

Collision of One-Dimensional Nonlinear Chains

We investigate one-dimensional collisions of unharmonic chains and a rigid wall. We find that the coefficient of restitution (COR) is strongly dependent on the velocity of colliding chains and has a minimum value at a certain velocity. The relationship between COR and collision velocity is derived for low-velocity collisions using perturbation methods. We found that the velocity dependence is characterized by the exponent of the lowest unharmonic term of interparticle potential energy

Scaling properties of granular materials

Given an assembly of viscoelastic spheres with certain material properties, we raise the question how the macroscopic properties of the assembly will change if all lengths of the system, i.e. radii, container size etc., are scaled by a constant. The result leads to a method to scale down experiments to lab-size.Comment: 4 pages, 2 figure

The impact of two-dimensional elastic disk

The impact of a two-dimensional elastic disk with a wall is numerically studied. It is clarified that the coefficient of restitution (COR) decreases with the impact velocity. The result is not consistent with the recent quasi-static theory of inelastic collisions even for very slow impact. The abrupt drop of COR is found due to the plastic deformation of the disk, which is assisted by the initial internal motion.(to be published in J. Phys. Soc. Jpn.)Comment: 6 Pages,2 figure