Poisson-to-Wigner crossover transition in the nearest-neighbor spacing statistics of random points on fractals

We show that the nearest-neighbor spacing distribution for a model that consists of random points uniformly distributed on a self-similar fractal is the Brody distribution of random matrix theory. In the usual context of Hamiltonian systems, the Brody parameter does not have a definite physical meaning, but in the model considered here, the Brody parameter is actually the fractal dimension. Exploiting this result, we introduce a new model for a crossover transition between Poisson and Wigner statistics: random points on a continuous family of self-similar curves with fractal dimension between 1 and 2. The implications to quantum chaos are discussed, and a connection to conservative classical chaos is introduced.Comment: Low-resolution figure is included here. Full resolution image available (upon request) from the author

Zeta Function Zeros, Powers of Primes, and Quantum Chaos

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensities are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series. The connections to quantum chaos and semiclassical physics are discussed

Semiclassical Trace Formulas for Noninteracting Identical Particles

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families. The identical nature of the particles also introduces discrete permutational symmetries. We exploit the formalism of Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied semiclassical dynamics in the presence of continuous symmetries, to derive many-body trace formulas for the full and symmetry-reduced densities of states. Numerical studies of the three-particle cardioid billiard are used to explicitly illustrate and test the results of the theory.Comment: 29 pages, 11 figures, submitted to PR

Parametric linear vibration response studies for longitudinal whole-body gradient coils: Theoretical predictions based on an exact linear elastodynamic model

Passive reduction of gradient coil (GC) cylinder vibration depends critically on a thorough knowledge of how all pertinent physical parameters affect the vibration response. In this paper, we employ a recently introduced linear elastodynamic Z-coil model to study how the displacement response of a whole-body GC cylinder (subject to exclusive excitation of its Z-coil windings) is affected by independent regularized variations in its: (i) length; (ii) radial thickness; (iii) mass density; (iv) Poisson ratio; and (v) Young modulus (stiffness). The results exhibit a rich variety of behaviors at different excitation frequencies, and in the parameter ranges of interest, the displacement response is found to be particularly sensitive to variations in cylinder geometry and mass density. The results also show that, with the exception of the stiffness, there are no optimal ranges of regularized values of the considered parameters that will reduce the displacement (and hence the vibration) of a GC cylinder at all frequencies of interest. For typical GC cylinder geometries and densities, and under the condition that only the Z-coil windings are excited, the model predicts that increasing the cylinder stiffness above 100 GPa will reduce vibration at all frequencies below 2000 Hz

Parametric modeling of steady-state gradient coil vibration: Resonance dynamics under variations in cylinder geometry

Gradient coil (GC) vibration is the root cause of many problems in MRI adversely affecting scanner performance, image quality, and acoustic noise levels. A critical issue is that GC vibration will be significantly increased close to any GC mechanical resonances. It is well known that altering the dimensions of a GC fundamentally affects the mechanical resonances excited by the GC windings. The precise nature of the effects (i.e., how the resonances are affected) is however not well understood. The purpose of the present paper is to study how the mechanical resonances excited by closed whole-body Z-gradient coils are affected by variations in cylinder geometry. A mathematical Z-gradient coil vibration model recently developed and validated by the authors is used to theoretically study the resonance dynamics under variation(s) in cylinder: (i) length, (ii) mean radius, and (iii) radial thickness. The forced-vibration response to Lorentz-force excitation is in each case analyzed in terms of the frequency response of the GC cylinder\u27s displacement. In cases (i) and (ii), the qualitative dynamics are simple: reducing the cylinder length and/or mean radius causes all mechanical resonances to shift to higher frequencies. In case (iii), the qualitative dynamics are much more complicated with different resonances shifting in different directions and additional dependencies on the cylinder length. The more detailed dynamics are intricate owing to the fact that resonances shift at comparatively different rates and this leads to several novel and theoretically interesting predicted effects. Knowledge of these effects advance our understanding of the basic mechanics of GC vibration and offer practically useful insights into how such vibration may be passively reduced

Constructing separable non-2π-periodic solutions to the Navier-Lamé equation in cylindrical coordinates using the Buchwald representation: Theory and applications

In a previous paper (Adv. Appl. Math. Mech., 10 (2018), pp. 1025-1056), we used the Buchwald representation to construct several families of separable cylindrical solutions to the Navier-Lamé equation; these solutions had the property of being 2π-periodic in the circumferential coordinate. In this paper, we extend the analysis and obtain the complementary set of separable solutions whose circumferential parts are elementary 2π-aperiodic functions. Collectively, we construct eighteen distinct families of separable solutions; in each case, the circumferential part of the solution is one of three elementary 2π-aperiodic functions. These solutions are useful for solving a wide variety of dynamical problems that involve cylindrical geometries and for which 2π-periodicity in the angular coordinate is incompatible with the given boundary conditions. As illustrative examples, we show how the obtained solutions can be used to solve certain forced-vibration problems involving open cylindrical shells and open solid cylinders where (by virtue of the boundary conditions) 2π-periodicity in the angular coordinate is inappropriate. As an addendum to our prior work, we also include an illustrative example of a certain type of asymmetric problem that can be solved using the particular 2π-periodic subsolutions that ensue when there is no explicit dependence on the circumferential coordinate