Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure

Consider a mixed quantum mechanical state, describing a statistical ensemble in terms of an arbitrary density operator $\rho$ of low purity, \tr\rho^2\ll 1, and yielding the ensemble averaged expectation value \tr(\rho A) for any observable $A$. Assuming that the given statistical ensemble $\rho$ is generated by randomly sampling pure states $|\psi>$ according to the corresponding so-called Gaussian adjusted projected measure $[$Goldstein et al., J. Stat. Phys. 125, 1197 (2006)$]$, the expectation value $$ is shown to be extremely close to the ensemble average \tr(\rho A) for the overwhelming majority of pure states $|\psi>$ and any experimentally realistic observable $A$. In particular, such a `typicality' property holds whenever the Hilbert space \hr of the system contains a high dimensional subspace \hr_+\subset\hr with the property that all |\psi>\in\hr_+ are realized with equal probability and all other |\psi> \in\hr are excluded.Comment: accepted for publication in J. Stat. Phy

The Process of price formation and the skewness of asset returns

Distributions of assets returns exhibit a slight skewness. In this note we show that our model of endogenous price formation \cite{Reimann2006} creates an asymmetric return distribution if the price dynamics are a process in which consecutive trading periods are dependent from each other in the sense that opening prices equal closing prices of the former trading period. The corresponding parameter $\alpha$ is estimated from daily prices from 01/01/1999 - 12/31/2004 for 9 large indices. For the S&P 500, the skewness distribution of all its constituting assets is also calculated. The skewness distribution due to our model is compared with the distribution of the empirical skewness values of the ingle assets.Comment: 9 pages, 2 figure

"All You Need is Love" - and What about Gender? Engendering Burton's Human Needs Theory

Ye

Wavefunction localization and its semiclassical description in a 3-dimensional system with mixed classical dynamics

We discuss the localization of wavefunctions along planes containing the shortest periodic orbits in a three-dimensional billiard system with axial symmetry. This model mimicks the self-consistent mean field of a heavy nucleus at deformations that occur characteristically during the fission process [1,2]. Many actinide nuclei become unstable against left-right asymmetric deformations, which results in asymmetric fragment mass distributions. Recently we have shown [3,4] that the onset of this asymmetry can be explained in the semiclassical periodic orbit theory by a few short periodic orbits lying in planes perpendicular to the symmetry axis. Presently we show that these orbits are surrounded by small islands of stability in an otherwise chaotic phase space, and that the wavefunctions of the diabatic quantum states that are most sensitive to the left-right asymmetry have their extrema in the same planes. An EBK quantization of the classical motion near these planes reproduces the exact eigenenergies of the diabatic quantum states surprisingly well.Comment: 4 pages, 5 figures, contribution to the Nobel Symposium on Quantum Chao

Maximal Slicing for Puncture Evolutions of Schwarzschild and Reissner-Nordstr\"om Black Holes

We prove by explicit construction that there exists a maximal slicing of the Schwarzschild spacetime such that the lapse has zero gradient at the puncture. This boundary condition has been observed to hold in numerical evolutions, but in the past it was not clear whether the numerically obtained maximal slices exist analytically. We show that our analytical result agrees with numerical simulation. Given the analytical form for the lapse, we can derive that at late times the value of the lapse at the event horizon approaches the value ${3/16}\sqrt{3} \approx 0.3248$, justifying the numerical estimate of 0.3 that has been used for black hole excision in numerical simulations. We present our results for the non-extremal Reissner-Nordstr\"om metric, generalizing previous constructions of maximal slices.Comment: 21 pages, 9 figures, published version with changes to Sec. VI