A statistical law for multiplicities of SU(3) irreps (lambda, mu) in the plethysm {eta} circle times(3) {m} -> (lambda, mu)

A statistical law for the multiplicities of the SU(3) irreps (lambda, mu) in the reduction of totally symmetric irreducible representations {m} of U(N), N = (eta + 1) (eta + 2)/2 with eta being the three-dimensional oscillator major shell quantum number, is derived in terms of the quadratic and cubic invariants of SU(3), by determining the first three terms of an asymptotic expansion for the multiplicities. To this end, the bivariate Edgeworth expansion known in statistics is used. Simple formulae, in terms of m and eta, for all the parameters in the expansion are derived. Numerical tests with large m and eta = 4, 5 and 6 show good agreement with the statistical formula for the SU(3) multiplicities.

Spectral fluctuation and 1/f(alpha) noise in the energy level statistics of interacting trapped bosons

It has been recently shown numerically that the transition from integrability to chaos in quantum systems and the corresponding spectral fluctuations are characterized by 1/f(alpha) noise with 1 <= alpha <= 2. The system of interacting trapped bosons is inhomogeneous and complex. The presence of an external harmonic trap makes it more interesting as, in the atomic trap, the bosons occupy partly degenerate single-particle states. Earlier theoretical and experimental results show that at zero temperature the low-lying levels are of a collective nature and high-lying excitations are of a single-particle nature. We observe that for few bosons, the P(s) distribution shows the Shnirelman peak, which exhibits a large number of quasidegenerate states. For a large number of bosons the low-lying levels are strongly affected by the interatomic interaction, and the corresponding level fluctuation shows a transition to a Wigner distribution with an increase in particle number. It does not follow Gaussian orthogonal ensemble random matrix predictions. For high-lying levels we observe the uncorrelated Poisson distribution. Thus it may be a very realistic system to prove that 1/f(alpha) noise is ubiquitous in nature.

Energy-level statistics of interacting trapped bosons

It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s = 0, which signifies the presence of a large number of quasidegenerate states.

A Kinetostatic Formulation for Load-Flow Visualization in Compliant Mechanisms

Load flow visualization, which is an important step in structural and machine assembly design may aid in the analysis and eventual synthesis of compliant mechanisms. In this paper, we present a kineto-static formulation to visualize load flow in compliant mechanisms. This formulation uses the concept of transferred forces to quantify load flow from input to the output of a compliant mechanism. The magnitude and direction of load flow in the constituent members enables functional decomposition of the compliant mechanism into (i) Constraints (C): members that are constrained to deform in a particular direction and (ii) Transmitters (T): members that transmit load to the output. Furthermore, it is shown that a constraint member and an adjacent transmitter member can be grouped together to constitute a fundamental building block known as an CT set whose load flow behavior is maximally decoupled from the rest of the mechanism. We can thereby explain the deformation behavior of a number of compliant mechanisms from literature by visualizing load flow, and identifying building blocks.

A Metric to Evaluate and Synthesize Distributed Compliant Mechanisms

Compliant mechanisms with evenly distributed stresses have better load-bearing ability and larger range of motion than mechanisms with compliance and stresses lumped at flexural hinges. In this paper, we present a metric to quantify how uniformly the strain energy of deformation and thus the stresses are distributed throughout the mechanism topology. The resulting metric is used to optimize cross-sections of conceptual compliant topologies leading to designs with maximal stress distribution. This optimization framework is demonstrated for both single-port mechanisms and single-input single-output mechanisms. It is observed that the optimized designs have lower stresses than their nonoptimized counterparts, which implies an ability for single-port mechanisms to store larger strain energy, and single-input single-output mechanisms to perform larger output work before failure.