There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

This paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For $sl1$ the system is localized at zero temperature, and at finite $T$ relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For $sg2$ it undergoes underdamped coherent oscillations for all relevant temperatures, while for $1lsl2$ there is a crossover from coherent oscillation to overdamped relaxation as $T$ increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, $s=1$, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature $T$: over most of the ($\ensuremath{\alpha}$,$T$) plane (including the whole region $\ensuremath{\alpha}g1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low $T$ and $0l\ensuremath{\alpha}l\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.

/pdf/dynamics-of-the-dissipative-two-state-system-4kzh83zg4g.pdf

Dynamics of the dissipative two-state system

This paper has two purposes: 1) to clarify the relationship between the quantum theory of radiation, where the electromagnetic field-expansion coefficients satisfy commutation relations, and the semiclassical theory, where the electromagnetic field is considered as a definite function of time rather than as an operator; and 2) to apply some of the results in a study of amplitude and frequency stability in a molecular beam maser. In 1), it is shown that the semiclassical theory, when extended te take into account both the effect of the field on the molecules and the effect of the molecules on the field, reproduces almost quantitatively the same laws of energy exchange and coherence properties as the quantized field theory, even in the limit of one or a few quanta in the field mode. In particular, the semiclassical theory is shown to lead to a prediction of spontaneous emission, with the same decay rate as given by quantum electrodynamics, described by the Einstein A coefficients. In 2), the semiclassical theory is applied to the molecular beam maser. Equilibrium amplitude and frequency of oscillation are obtained for an arbitrary velocity distribution of focused molecules, generalizing the results obtained previously by Gordon, Zeiger, and Townes for a singel-velocity beam, and by Lamb and Helmer for a Maxwellian beam. A somewhat surprising result is obtained; which is that the measurable properties of the maser, such as starting current, effective molecular Q, etc., depend mostly on the slowest 5 to 10 per cent of the molecules. Next we calculate the effect of amplitude and frequency of oscillation, of small systematic perturbations. We obtain a prediction that stability can be improved by adjusting the system so that the molecules emit all their energy h Ω to the field, then reabsorb part of it, before leaving the cavity. In general, the most stable operation is obtained when the molecules are in the process of absorbing energy from the radiation as they leave the cavity, most unstable when they are still emitting energy at that time. Finally, we consider the response of an oscillating maser to randomly time-varying perturbations. Graphs are given showing predicted response to a small superimposed signal of a frequency near the oscillation frequency. The existence of "noise enhancing" and "noise quieting" modes of operation found here is a general property of any oscillating system in which amplitude is limited by nonlinearity.

Comparison of quantum and semiclassical radiation theories with application to the beam maser

The Theory of a general quantum system interacting with a linear dissipative system

A simple, rigorous geometrical representation for the Schrodinger equation is developed to describe the behavior of an ensemble of two quantum‐level, noninteracting systems which are under the influence of a perturbation. In this case the Schrodinger equation may be written, after a suitable transformation, in the form of the real three‐dimensional vector equation dr/dt=ω×r, where the components of the vector r uniquely determine ψ of a given system and the components of ω represent the perturbation. When magnetic interaction with a spin ½ system is under consideration, ``r'' space reduces to physical space. By analogy the techniques developed for analyzing the magnetic resonance precession model can be adapted for use in any two‐level problems. The quantum‐mechanical behavior of the state of a system under various different conditions is easily visualized by simply observing how r varies under the action of different types of ω. Such a picture can be used to advantage in analyzing various MASER‐type devi...

Geometrical Representation of the Schrödinger Equation for Solving Maser Problems

Spectral estimation procedures which employ several prolate spheroidal sequences as tapers have been shown to yield better results than standard single-taper spectral analysis when used on a variety of engineering data. We apply the adaptive multitaper spectral estimation method of Thomson (1982) to a number of high-resolution digital seismic records and compare the results to those obtained using standard single-taper spectral estimates. Single-taper smoothed-spectrum estimates are plagued by a trade-off between the variance of the estimate and the bias caused by spectral leakage. Applying a taper to reduce bias discards data, increasing the variance of the estimate. Using a taper also unevenly samples the record. Throwing out data from the ends of the record can result in a spectral estimate which does not adequately represent the character of the spectrum of nonstationary processes like seismic waveforms. For example, a discrete Fourier transform of an untapered record (i.e., using a boxcar taper) produces a reasonable spectral estimate of the large-amplitude portion of the seismic source spectrum but cannot be trusted to provide a good estimate of the high-frequency roll-off. A discrete Fourier transform of the record multiplied by a more severe taper (like the Hann taper) which is resistant to spectral leakage leads to a reliable estimate of high-frequency spectral roll-off, but this estimate weights the analyzed data unequally. Therefore single-taper estimators which are less affected by leakage not only have increased variance but also can misrepresent the spectra of nonstationary data. The adaptive multitaper algorithm automatically adjusts between these extremes. We demonstrate its advantages using 16-bit seismic data recorded by instruments in the Anza Telemetered Seismic Network. We also present an analysis demonstrating the superiority of the multitaper algorithm in providing low-variance spectral estimates with good leakage resistance which do not overemphasize the central portion of the record.

Multitaper spectral analysis of high-frequency seismograms

http://wwwprof.uniandes.edu.co/~gprieto/publication/gprieto_mtspec.pdf

A Fortran 90 library for multitaper spectrum analysis

The large volume of broadband waveforms that are being acquired by USArray (http://www.iris.edu/USArray/), the seismology component of the national Earth science program EarthScope (http://www.earthscope.org/), offer unique opportunities for seismic imaging. Constraining structures on a range of length scales and understanding their physical and chemical causes is a prerequisite for understanding the relationship between near surface and deeper mantle processes. One can expect that, eventually, full wave tomography ( e.g. , De Hoop and van der Hilst 2005; De Hoop et al. 2006; Tromp et al. 2005; Zhao and Jordan 2006) with broadband USArray waveforms will produce superior insight into the structure of the mantle beneath North America, but linearized tomographic inversion of phase arrival time data readily yields exciting results in regions where data from dense seismograph networks is available. We will use travel times from the transportable component of USArray, hereinafter referred to as USArrayTA, and from other sources, such as the EHB data base (Engdahl et al. 1998), to constrain 3-D mantle heterogeneity beneath North America.

Tomographic images based on USArrayTA data will help us understand first-order geological structure of and processes in the mantle beneath North America. Examples include, but are not restricted to: 1) the transition from the stable continental lithosphere at the center of the North American continent to the tectonically active domains farther west, 2) the Cascadian subduction system, 3) the Yellowstone hotspot, and 4) the relationship between current and past episodes of subduction and upper mantle upwellings and processes deeper in the mantle. Before the advent of USArrayTA, insight into mantle structure beneath the western United States was obtained either from pieced-together regional P -wave studies ( e.g. , figure 1A, after Dueker et al. 2001) or from global travel time or surface wave tomography (figures 1B, C, and D, …

Frank L. Vernon

Papers

The Theory of a general quantum system interacting with a linear dissipative system

Geometrical Representation of the Schrödinger Equation for Solving Maser Problems

Multitaper spectral analysis of high-frequency seismograms

A Fortran 90 library for multitaper spectrum analysis

Upper Mantle Heterogeneity beneath North America from Travel Time Tomography with Global and USArray Transportable Array Data