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's 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.

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.

Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics.

Two decades ago, Wang and Ong, [Phys. Rev. A 55, 1522 (1997)]10.1103/PhysRevA.55.1522 hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this Rapid Communication, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically computed local box-counting dimension.

Wigner surmises and the two-dimensional homogeneous Poisson point process.

We derive a set of identities that relate the higher-order interpoint spacing statistics of the two-dimensional homogeneous Poisson point process to the Wigner surmises for the higher-order spacing distributions of eigenvalues from the three classical random matrix ensembles. We also report a remarkable identity that equates the second-nearest-neighbor spacing statistics of the points of the Poisson process and the nearest-neighbor spacing statistics of complex eigenvalues from Ginibre's ensemble of 2 x 2 complex non-Hermitian random matrices.

Spacing distributions for point processes on a regular fractal.

The homogeneous Poisson point process in Rd (denoted by Pd) is a basic model of stochastic geometry and modern statistical physics. Using ideas from fractal geometry, geometrical statistics, and random matrix theory, we introduce the model of random points on a self-similar fractal as a model of intermediate statistics, in the sense that the interpoint spacing statistics of the model are intermediate between those of P1 and P2 when the fractal dimension is in between 1 and 2, and intermediate between those of P2 and P3 when the fractal dimension is in between 2 and 3, and so on. We also introduce the idea of using a continuous family of such models to interpolate between P1 and P2 and thereby effectuate crossover transitions between P1 statistics and P2 statistics. We first derive the kth-nearest-neighbor spacing distribution for the general model, and then study the interpoint spacing statistics of several realizations of the model involving Sierpinski fractals in R2 and R3. We also study a realization of a continuous interpolation between P1 and P2, in particular a continuous interpolation between a point process on a line and a point process on a plane-filling curve, using the continuous family of self-similar Koch curves in R2. In the latter study, we specifically analyze the second-nearest-neighbor interpoint spacing statistics, which undergo a crossover transition between semi-Poisson and Ginibre statistics.

Poisson-to-Wigner crossover transition in the nearest-neighbor 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 dimensions between 1 and 2. The implications to quantum chaos are discussed, and a connection to conservative classical chaos is introduced.

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 integer powers. The formula consists 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 integer 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.