Quantum anti-zeno paradox

We establish an exact differential equation for the operator describing time-dependent measurements continuous in time and obtain a series solution. Suppose the projection operator E(t) = U(t)EUdagger(t) is measured continuously from t = 0 to T, where E is a projector leaving the initial state unchanged and U(t) a unitary operator obeying U(0) = 1. We prove that the probability of always finding E(t) = 1 from t = 0 to T is unity. If U(t) not equal1, the watched kettle is sure to "boil."     

Probability distribution of the maximum of a smooth temporal signal

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated with the probability that the process remains below a nonzero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=X(0)X(t).     

Lévy-Process Intrinsic Statistical Solutions of a Randomly Forced Burgers Equation

We consider Burgers equation forced by a brownian in space and white noise in time process \(\partial_{t}u+\frac{1}{2}\partial_{x}(u)^{2}=f(x,t)\), with \(E(f(x,t)f(y,s))=\frac{1}{2}(|x|+|y|-|x-y|)\*\delta(t-s)\) and we show that there exist intrinsic statistical solutions that are Lévy processes at any given positive time. We give the evolution equation for the characteristic exponent of such solutions; in particular we give the explicit solution in the case u ₀(x)=0.     

Directed current due to broken time-space symmetry

We consider the classical dynamics of a particle in a one-dimensional space-periodic potential U(X) = U(X+2pi) under the influence of a time-periodic space-homogeneous external field E(t) = E(t+T). If E(t) is neither a symmetric function of t nor antisymmetric under time shifts E(t+/-T/2) not equal-E(t), an ensemble of trajectories with zero current at t = 0 yields a nonzero finite current as t-->infinity. We explain this effect using symmetry considerations and perturbation theory. Finally we add dissipation (friction) and demonstrate that the resulting set of attractors keeps the broken symmetry property in the basins of attraction and leads to directed currents as well.     

Remark on the (non)convergence of ensemble densities in dynamical systems

We consider a dynamical system with state space M, a smooth, compact subset of some R(n), and evolution given by T(t), x(t)=T(t)x, x in M; T(t) is invertible and the time t may be discrete, t in Z, T(t)=T(t), or continuous, t in R. Here we show that starting with a continuous positive initial probability density rho(x,0)>0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.     

Average shape of a fluctuation: universality in excursions of stochastic processes

We study the average shape of a fluctuation of a time series x(t), which is the average value (T) before x(t) first returns at time T to its initial value x(0). For large classes of stochastic processes, we find that a scaling law of the form (T) = T(alpha)f(t/T) is obeyed. The scaling function f(s) is, to a large extent, independent of the details of the single increment distribution, while it encodes relevant statistical information on the presence and nature of temporal correlations in the process. We discuss the relevance of these results for Barkhausen noise in magnetic systems.     

Stability and equilibrium states of infinite classical systems

We prove that any stationary state describing an infinite classical system which is stable under local perturbations (and possesses some strong time clustering properties) must satisfy the classical KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (c) 1997 American Institute of Physics.     

Photon transitions in Upsilon(2S) and Upsilon(3S) decays

We have studied the inclusive photon spectra in Upsilon(2S) and Upsilon(3S) decays using a large statistics data sample obtained with the CLEO III detector. We present the most precise measurements of electric dipole (E1) photon transition rates and photon energies for Upsilon(2S) --> gammachi(bJ)(1P) and Upsilon(3S) --> gammachi(bJ)(2P) (J = 0, 1, 2). We measure the rate for a rare E1 transition Upsilon(3S) --> gammachi(b0)(1P) for the first time. We also set upper limits on the rates for the hindered magnetic dipole (M1) transitions to the eta(b)(1S) and eta(b)(2S) states.     

Shape oscillations and stability of charged microdroplets

Classical Rayleigh theory predicts an instability of a surface charged liquid sphere, when the Coulomb energy E(C) exceeds twice the surface energy E(S). Previously, electrified liquid droplets have been found to disintegrate at a fissility X=E(C)/2E(S) well below one, however. We determine the stability of charged droplets in an electrodynamic levitator by observing the amplitude and phase of their quadrupolar shape oscillations as a function of the fissility. With this novel approach, which does not rely on an independent determination of the charge and surface tension of the droplets, we are able to confirm for the first time the Rayleigh limit of stability at X=1 for micrometer sized droplets of ethylene glycol.     

Entropy dissipation and moment production for the Boltzmann equation

LetH(f/M)=flog(f/M)dv be the relative entropy off and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation byD(f)=Q(f,f) logf dv. An example is presented which shows that |D(f)/H(f/M)| can be arbitrarily small. This example is a sequence of isotropic functions, and the estimates are very explicitly given by a simple formula forD which holds for such functions. The paper also gives a simplified proof of the so-called Povzner inequality, which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision. That inequality is then used to prove that f(v) |v|^s dt<C(t), wheref is the solution of the spatially homogeneous Boltzmann equation. HereC(t) is an explicitly given function dependings and the mass, energy, and entropy of the initial data.     

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

Orientational dependence of luminescence yield excited by fast electrons in NaCl crystals

An investigation is made of the orientational dependence of the integral luminescence yield when NaCl(T1) single crystals are irradiated by electrons of energy 0.8–2.5 MeV in the direction 100. It is shown that the luminescence yield at small excitation densities depends linearly on the specific ionizational energy losses of the bombarding particles. The ionizational energy losses are calculated for the case when electrons of energy E=1.0 MeV pass through an NaCl single crystal in the direction 100. The minimum observed in the luminescence yield is due to electron capture in canalization conditions and decrease in energy losses. The maximum in the calculated curve at small angles of disorientation is due to the focusing of fast electrons by atomic chains.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 75–79, March, 1979.It remains to thank A. M. Taratin for performing the computer calculations and S. A. Vorob'ev for supervising the work.     

Wigner Kirkwood ℏ-expansion of the density matrix in inhomogeneous superfluid Fermi systems

The generalized density matrix of superfluid inhomogeneous Fermi systems is expanded in powers of up to order ². This constitutes the generalisation of the Wigner Kirkwood -expansion of the density matrix of normal fluid systems to the pairing case.One of the authors (P.S.) is very grateful to D. Gogny for contributions in an early stage of this work several years back. He also acknowledges fruitful discussions with M. Centelles and X. Vinas.     

Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

If , and is a finite (nonabelian) group, then is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of . We characterize when MCA are group endomorphisms of , and show that MCA on inherit a natural structure theory from the structure of . We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Random Motions at Finite Speed in Higher Dimensions

We present a general method of studying the transport process , t≥0, in the Euclidean space ℝ ^m , m≥2, based on the analysis of the integral transforms of its distributions. We show that the joint characteristic functions of are connected with each other by a convolution-type recurrent relation. This enables us to prove that the characteristic function (Fourier transform) of in any dimension m≥2 satisfies a convolution-type Volterra integral equation of second kind. We give its solution and obtain the characteristic function of in terms of the multiple convolutions of the kernel of the equation with itself. An explicit form of the Laplace transform of the characteristic function in any dimension is given. The complete solution of the problem of finding the initial conditions for the governing partial differential equations, is given. We also show that, under the standard Kac condition on the speed of the motion and on the intensity of the switching Poisson process, the transition density of the isotropic transport process converges to the transition density of the m-dimensional homogeneous Brownian motion with zero drift and diffusion coefficient depending on the dimension m. We give the conditional characteristic functions of the isotropic transport process in terms of the inverse Laplace transform of the powers of the Gauss hypergeometric function. Some important models of the isotropic transport processes in lower dimensions are considered and some known results are derived as the particular cases of our general model by means of the method developed.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.     

Superconducting terminals as sensitive probes for scarred states

When a quantum-chaotic normal conductor is coupled to a superconductor, the random-matrix theory (RMT) predicts that a gap opens up in the excitation spectrum which is of universal size E(g)(RMT) approximately 0.3 Planck/t(D), where t(D) is the mean scattering time between Andreev reflections. We show that a scarred state of long lifetime t(S)>t(D) suppresses the excitation gap over a window DeltaE approximately 2E(g)(RMT) which can be much larger than the narrow resonance width GammaS= Planck/t(S) of the scar in the normal system. The minimal value of the excitation gap within this window is given by GammaS/2相似文献