Periodic orbit quantization of the Sinai billiard in the small scatterer limit

We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared to the wave length 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact KKR determinant, which we generalize here for the A_1 subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus infinity within the range of validity of the diffractive approximation kR <<4. We study the statistics exhibited by spectra for fixed values of c. It is Poissonian for |c|=infinity, provided the disk is placed inside a rectangle whose sides obeys some constraints. For c=0 we find a good agreement of the level spacing distribution with GOE, whereas the form factor and two-point correlation function are similar but exhibit larger deviations. By varying the parameter c from 0 to infinity the level statistics interpolates smoothly between these limiting cases.Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math. Ge

On the duality between periodic orbit statistics and quantum level statistics

We discuss consequences of a recent observation that the sequence of periodic orbits in a chaotic billiard behaves like a poissonian stochastic process on small scales. This enables the semiclassical form factor $K_{sc}(\tau)$ to agree with predictions of random matrix theories for other than infinitesimal $\tau$ in the semiclassical limit.Comment: 8 pages LaTe

Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions

We compute the Lyapunov exponent, generalized Lyapunov exponents and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than periodic orbits, a re used in order to avoid the convergence problems of cycle expansions. The emphasis is on the relation between the analytic structure of the zeta function, where a branch cut plays an important role, and the asymptotic dynamics of the system. We find a diverging diffusion constant $D(t) \sim \log t$ and a phase transition for the generalized Lyapunov exponents.Comment: 14 pages LaTeX, figs 2-3 on .uu file, fig 1 available from autho

Sub‐phenotyping Metabolic Disorders Using Body Composition: An Individualized, Nonparametric Approach Utilizing Large Data Sets

Objective This study performed individual-centric, data-driven calculations of propensity for coronary heart disease (CHD) and type 2 diabetes (T2D), utilizing magnetic resonance imaging-acquired body composition measurements, for sub-phenotyping of obesity and nonalcoholic fatty liver disease (NAFLD). Methods A total of 10,019 participants from the UK Biobank imaging substudy were included and analyzed for visceral and abdominal subcutaneous adipose tissue, muscle fat infiltration, and liver fat. An adaption of the k-nearest neighbors algorithm was applied to the imaging variable space to calculate individualized CHD and T2D propensity and explore metabolic sub-phenotyping within obesity and NAFLD. Results The ranges of CHD and T2D propensity for the whole cohort were 1.3% to 58.0% and 0.6% to 42.0%, respectively. The diagnostic performance, area under the receiver operating characteristic curve (95% CI), using disease propensities for CHD and T2D detection was 0.75 (0.73-0.77) and 0.79 (0.77-0.81). Exploring individualized disease propensity, CHD phenotypes, T2D phenotypes, comorbid phenotypes, and metabolically healthy phenotypes were found within obesity and NAFLD. Conclusions The adaptive k-nearest neighbors algorithm allowed an individual-centric assessment of each individual’s metabolic phenotype moving beyond discrete categorizations of body composition. Within obesity and NAFLD, this may help in identifying which comorbidities a patient may develop and consequently enable optimization of treatment

Spectral statistics in chaotic systems with a point interaction

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(tau) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order tau^2 and tau^3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde

Stability ordering of cycle expansions

We propose that cycle expansions be ordered with respect to stability rather than orbit length for many chaotic systems, particularly those exhibiting crises. This is illustrated with the strong field Lorentz gas, where we obtain significant improvements over traditional approaches.Comment: Revtex, 5 incorporated figures, total size 200

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-via-canonicity. Specifically, we use the topological theory of canonical (in) equations in distributive lattice expansions to show that distributive substructural logics are strongly complete with respect to their relational semantics. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions

Geometrical theory of diffraction and spectral statistics

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit $\hbar \to 0$. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.

Symmetry Decomposition of Potentials with Channels

We discuss the symmetry decomposition of the average density of states for the two dimensional potential $V=x^2y^2$ and its three dimensional generalisation $V=x^2y^2+y^2z^2+z^2x^2$. In both problems, the energetically accessible phase space is non-compact due to the existence of infinite channels along the axes. It is known that in two dimensions the phase space volume is infinite in these channels thus yielding non-standard forms for the average density of states. Here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels, leading to terms which are essentially leading order. We verify these results numerically and also observe a peculiar numerical effect which we associate with the channels. In three dimensions, the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials --- however there are still non-generic effects related to some of the group elements