Spatial correlation function time dependent

The spatial correlation approach is readily extended to the nonstationary case by defining time-dependent (classical and quantum) spatial correlation function and correlation length that is, one applies Eqs. (4.2) and (4.3) to time-dependent densities. An example, in which the time dependence of... 

The electron-spin time-correlation functions of Eq. (56) were evaluated numerically by constructing an ensemble of trajectories containing the time dependence of the spin operators and spatial functions, in a manner independent of the validity of the Redfield limit for the rotational modulation of the static ZFS. Before inserting thus obtained electron-spin time-correlation functions into an equation closely related to Eq. (38), Abernathy and Sharp also discussed the effect of distortional/vibrational processes on the electron spin relaxation. They suggested that the electron spin relaxation could be described in terms of simple exponential decay rate constant Ts, expressed as a sum of a rotational and a distortional contribution ... 

The phase-dependent directionality of photocurrents produced by such a detector entails advantageous properties of the photocurrents cross correlations in nonoverlapping time intervals or spatial regions (considered in Section 4.2.2). These directional time-dependent correlations are measured with one detector only. They involve solely terms dependent on LO phases, in contrast to similar correlations measured by conventional photocounters, which inevitably contain terms depending on photon fluxes such as the LO excess noise. Owing to these properties, the mean autocorrelation function of the SL quadrature is shown in the schemes considered here to be measurable without terms related to the LO noise. LO shot noise, which affects the degree of accuracy to which this autocorrelation is measured (i.e., its variance) is easily obtainable from zero time delay correlations because the LO excess noise is suppressed. The combined measurements of cross correlations and zero time delay correlations yield complete information on the SL in these schemes. 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

Deviation from standard chemical kinetics described in (Section 2.1.1) can happen only if the reaction rate K (t) reveals its own non-monotonous time dependence. Since K(t) is a functional of the correlation functions, it means that these functions have to possess their own motion, practically independent on the time development of concentrations. The correlation functions characterize the intermediate order in the particle distribution in a spatially-homogeneous system. Change of such an intermediate order could be interpreted as a series of structural transitions. 

Spectra of other dynamical variables like CoM velocities and zeolite window diameters can also be obtained by Fourier transformation of the appropriate time correlation functions. Calculation of spectra for different spatial components of a time dependent quantity provides useful information about the anisotropy of the corresponding motion. 

Condensed matter phases and structures are commonly reached via symmetry breaking transitions. In such systems, when the continuous symmetry is broken, temporary domain-t5q)e patterns are formed. The domain structures eventually coarsen, and disappear in the long-time limit, leaving a uniform broken-symmetry state. This state possesses so-called long-range order (LRO), in which the spatially dependent order parameter correlation function does not decay to zero in the limit of large distances. 

A detailed description of the time evolution of spatial correlations in liquids requires the introduction of a time-dependent generalization of the radial distribution function. It is the van Hove correlation function [24] which retains the microscopic nature of the system and yet are tractable within the current development in the statistical mechanical theory of liquids. 

It is thus necessary to use the correlation technique. In our horaodyne correlation experiment, we directly obtain C(t)"a(l+b02(t)), in which b is a spatial coherence factor and a depends on the average number of photocounts in the sampling time. The 0(t) correlation function is the Fourier Transform of the central line of Fig. 1. This correlation function will be supposed latter to have a particular form depending on two parameters characterizing the relaxation process. 

G(r, t) - space- and time-dependent pair correlation function = Debye-Waller temperature factor Ge(r) = equilibrium spatial pair cor relation function for atoms Go(r) == instantaneous spatial pair correlation function... 

Summary of Transducer Comparison. In summary, the point measurement techniques of CTA and LDA can offer good spatial and temporal response. This makes them ideal for measurements of both time-independent flow statistics, such as moments of velocity (mean, RMS, etc.) and time-dependent flow statistics such as flow spectra and correlation functions at a point. Although rakes of these sensors can be built, multipoint measurements are limited due primarily to cost. 

The resolution of the time delay measurement, and therefore the velocity resolution, will depend on the spread in the correlation function which in turn will depend on (i) the range of particle speeds coupled with the distance between the detectors and (ii) the spatial (hence time) resolution of the detectors. In the case of capacitance sensors, the spatial resolution will be approximately the length of the electrodes in the direction of flow, whereas in a radiation absorption technique the width of the received beam will be the determining factor. The spatial resolution of pressure and acoustic signals is much longer as the acoustic transmission properties of the gas and (in the case of acoustic techniques) the pipe is relevant. 

The time dependence of the field correlation function is therefore determined by the time correlation function of a spatial Fourier component of the concentration, namely... 

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