A theoretical development is presented which results in a relationship between the expectation value of the standard deviation of the frequency fluctuations for any finite number of data samples and the infinite time average value of the standard deviation, which provides an invariant measure of an important quality factor of a frequency standard. A practical and straightforward method of determining the power spectral density of the frequency fluctuations from the variance of the frequency fluctuations, the sampling time, the number of samples taken, and the dependence on system bandwidth is also developed. Additional insight is also given into some of the problems that arise from the presence of "flicker noise" (spectrum proportional to |ω|-1) modulation of the frequency of an oscillator. The theory is applied in classifying the types of noise on the signals of frequency standards made available at NBS, Boulder Laboratories, such as: masers (both H and N15H 3 ), the cesium beam frequency standard employed as the U. S. Frequency Standard, and rubidium gas cells. "Flicker noise" frequency modulation was not observed on the signals of masers for sampling times ranging from 0.1 second to 4 hours. In a comparison between the NBS hydrogen maser and the NBS III cesium beam, uncorrelated random noise was observed on the frequency fluctuations for sampling times extending to 4 hours; the fractional standard deviations of the frequency fluctuations were as low as 5 parts in 1014.

Statistics of atomic frequency standards

Elementary parameter‐differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker‐Campbell‐Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equationdY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated‐commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed‐form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed‐form single‐exponential solutions to the differential equationdY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple‐exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincare which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincare formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic‐oscillator canonical‐ensemble averages of general operators defined by the Weyl prescription. Solutions of the density‐matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.

Exponential Operators and Parameter Differentiation in Quantum Physics

The polarization of graphene is calculated exactly within the random phase approximation for arbitrary frequency, wavevector and doping. At finite doping, the static susceptibility saturates to a constant value for low momenta. At q = 2kF it has a discontinuity only in the second derivative. In the presence of a charged impurity this results in Friedel oscillations which decay with the same power law as the Thomas–Fermi contribution, the latter being always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization for low q and arbitrary ω is employed to calculate the dispersion relation and the decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations.

https://iopscience.iop.org/article/10.1088/1367-2630/8/12/318/pdf

Dynamical polarization of graphene at finite doping

http://staff.mechmining.uq.edu.au/klimenko/pub/pdf/Prog_Energy_Combust_Sci_1999_p.pdf

Conditional moment closure for turbulent combustion

A tutorial review of some time-domain methods of characterizing the performance of precision clocks and oscillators is presented. Characterizing both the systematic and random deviations is considered. The Allan variance and the modified Allan variance are defined, and methods of utilizing them are presented along with ranges and areas of applicability. The standa,rd deviation is contrasted and shoun not to be. in general. a good measure for precision clocks and oscillators. Once a proper characterization model has been developed, then optimum estimation and prediction techniques can be employed. Some important cases are illustrated. As precision clocks and oscillators become increasingly important in society. communication of their characteristics and specifications among the vendors, manufacturers. design engineers. managers, and metrologists of this equipment becomes increasingI> important.

/pdf/time-and-frequency-time-domain-characterization-estimation-2pv6zqwnx3.pdf

Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators

1. Introduction 2. The theory of generalised functions and their Fourier transforms 3. Definitions, properties and Fourier transforms of particular generalised functions 4. The asymptotic estimation of Fourier transforms 5. Fourier series Index.

R. A. Kunze

Papers

An Introduction to Fourier Analysis and Generalized Functions.