On the propagation of semiclassical Wigner functions

We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we re-discuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centered on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.Comment: 9 pages, 1 figure. To appear in J. Phys. A. This version matches the one set to print and differs from the previous one (07 Nov 2001) by the addition of two references, a few extra words of explanation and an augmented figure captio

Uniform approximation for the overlap caustic of a quantum state with its translations

The semiclassical Wigner function for a Bohr-quantized energy eigenstate is known to have a caustic along the corresponding classical closed phase space curve in the case of a single degree of freedom. Its Fourier transform, the semiclassical chord function, also has a caustic along the conjugate curve defined as the locus of diameters, i.e. the maximal chords of the original curve. If the latter is convex, so is its conjugate, resulting in a simple fold caustic. The uniform approximation through this caustic, that is here derived, describes the transition undergone by the overlap of the state with its translation, from an oscillatory regime for small chords, to evanescent overlaps, rising to a maximum near the caustic. The diameter-caustic for the Wigner function is also treated.Comment: 14 pages, 9 figure

Local quantum ergodic conjecture

The Quantum Ergodic Conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a delta-function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple observables, in agreement with Shnirelman's theorem, but this putative Wigner function violates several important requirements. Consequently, we transfer the conjecture to the Fourier transform of the Wigner function, that is, the chord function. We show that all the relevant consequences of the usual conjecture require only information contained within a small (Planck) volume around the origin of the phase space of chords: translations in ordinary phase space. Loci of complete orthogonality between a given eigenstate and its nearby translation are quite elusive for the Wigner function, but our local conjecture stipulates that their pattern should be universal for ergodic eigenstates of the same Hamiltonian lying within a classically narrow energy range. Our findings are supported by numerical evidence in a Hamiltonian exhibiting soft chaos. Heavily scarred eigenstates are remarkable counter-examples of the ergodic universal pattern.Comment: 4 figure

A percolation system with extremely long range connections and node dilution

We study the very long-range bond-percolation problem on a linear chain with both sites and bonds dilution. Very long range means that the probability $p_{ij}$ for a connection between two occupied sites $i,j$ at a distance $r_{ij}$ decays as a power law, i.e. $p_{ij} = \rho/[r_{ij}^\alpha N^{1-\alpha}]$ when $0 \le \alpha < 1$, and $p_{ij} = \rho/[r_{ij} \ln(N)]$ when $\alpha = 1$. Site dilution means that the occupancy probability of a site is $0 < p_s \le 1$. The behavior of this model results from the competition between long-range connectivity, which enhances the percolation, and site dilution, which weakens percolation. The case $\alpha=0$ with $p_s =1$ is well-known, being the exactly solvable mean-field model. The percolation order parameter $P_\infty$ is investigated numerically for different values of $\alpha$, $p_s$ and $\rho$. We show that in the ranges $0 \le \alpha \le 1$ and $0 < p_s \le 1$ the percolation order parameter $P_\infty$ depends only on the average connectivity $\gamma$ of sites, which can be explicitly computed in terms of the three parameters $\alpha$, $p_s$ and $\rho$

Scaling in a continuous time model for biological aging

In this paper we consider a generalization to the asexual version of the Penna model for biological aging, where we take a continuous time limit. The genotype associated to each individual is an interval of real numbers over which Dirac $\delta$--functions are defined, representing genetically programmed diseases to be switched on at defined ages of the individual life. We discuss two different continuous limits for the evolution equation and two different mutation protocols, to be implemented during reproduction. Exact stationary solutions are obtained and scaling properties are discussed.Comment: 10 pages, 6 figure

A Farm Level Analysis of the Economic Impact of the MARENA Program in Honduras

Impact Evaluation, Propensity Score Matching, Fixed Effects, Internal Rate of Return, Technical Efficiency, Honduras, International Development,

Orbit bifurcations and the scarring of wavefunctions

We extend the semiclassical theory of scarring of quantum eigenfunctions psi_{n}(q) by classical periodic orbits to include situations where these orbits undergo generic bifurcations. It is shown that |psi_{n}(q)|^{2}, averaged locally with respect to position q and the energy spectrum E_{n}, has structure around bifurcating periodic orbits with an amplitude and length-scale whose hbar-dependence is determined by the bifurcation in question. Specifically, the amplitude scales as hbar^{alpha} and the length-scale as hbar^{w}, and values of the scar exponents, alpha and w, are computed for a variety of generic bifurcations. In each case, the scars are semiclassically wider than those associated with isolated and unstable periodic orbits; moreover, their amplitude is at least as large, and in most cases larger. In this sense, bifurcations may be said to give rise to superscars. The competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed. We argue that there is a resulting universal hbar-scaling in the semiclassical asymptotics of these moments for irregular states in systems with a mixed phase-space dynamics. Finally, a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps.Comment: 24 pages, 6 Postscript figures, corrected some typo