A model of quantum reduction with decoherence

The problem of reduction (wave packet reduction) is reexamined under two simple conditions: Reduction is a last step completing decoherence. It acts in commonplace circumstances and should be therefore compatible with the mathematical frame of quantum field theory and the standard model. These conditions lead to an essentially unique model for reduction. Consistency with renormalization and time-reversal violation suggest however a primary action in the vicinity of Planck's length. The inclusion of quantum gravity and the uniqueness of space-time point moreover to generalized quantum theory, first proposed by Gell-Mann and Hartle, as a convenient framework for developing this model into a more complete theory.Comment: 20 pages. To be published in Physical Review

Effective power-law dependence of Lyapunov exponents on the central mass in galaxies

Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation $L\propto m^p$ between the mean Lyapunov exponent $L$ of stellar orbits chaotically scattered by a supermassive black hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the mass of the black hole over the mass of the galaxy. The exponent $p$ is found numerically to obtain values in the range $p \approx 0.3$--$0.5$. We propose a theoretical interpretation of these exponents, based on estimates of local `stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the black hole's sphere of influence. We thus predict $p=2/3-q$ with $q\approx 0.1$--$0.2$. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power law scaling of $L$ with $m$ holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the black hole affects mainly populations of thick tube-like orbits surrounding some low-order branches of the $x_1$ family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the Inner Lindbland resonance and the 4/1 resonance. Implications of the correlations between $L$ and $m$ to determining the rate of secular evolution of galaxies are discussed.Comment: 27 pages, 19 figure

A Central Limit Theorem for biased random walks on Galton-Watson trees

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$ offspring, moves closer to the root with probability $\lambda/(\lambda+d_v)$, and moves to each of the offspring with probability $1/(\lambda+d_v)$. It is known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$ (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $0<\lambda<m$ and $\v=0$ for $\lambda \ge m$. For all $\lambda \le m$, we prove a quenched CLT for |X_n|-n\v. (For $\lambda>m$ the walk is positive recurrent, and there is no CLT.) The most interesting case by far is $\lambda=m$, where the CLT has the following form: for almost every ${\cal T}$, the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle (previously, such a measure was explicitly known only for $\lambda=1$) and the construction of appropriate harmonic coordinates.Comment: 34 pages, 4 figure