Auto correlation

Auto-correlation and Inter-correlation Functions allow a good discrimination between these two types of defects by quantifying the resemblance between the different echoes and their derivatives. 

This function measures the correlation of the property A (it) to itself at two different times separated by the time interval t, averaged over the whole trajectory. The auto-correlation function is reversible in time [i.e., CaU) = it)], and it is stationary (i.e., (A(it + t)A(t))... 

FIG. 13 Scaling plot of the auto-correlation function (i EE(0) different temperatures for a bond fluctuation MC simulation [47]. 

Fig. 2.41 Auto-correlation functions for pressure and temperature fluctuations. Reprinted from Hetsroni et al. (2002b) with permission...

The auto-correlation functions for the pressure and temperature Rjj fluctuations are presented in Fig. 2.41. It is clear that the temporal behavior of the temperature fluctuations corresponds to that of the pressure fluctuations (Hetsroni et al. 2002b). 

Thermal resistance, gas constant, auto-correlation function, radius... 

Qualitatively, xc can be viewed as the time necessary for a reorientation by one radian. xc is very weak (10 11-10 12 s) for small size molecules in non-viscous solvents. Conversely, for large molecules (such as proteins in aqueous solution), it can reach much more important values (10 9 s or higher). All (normalized) auto-correlation spectral densities have the same expressions since, in the molecule, all directions are equivalent... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

A novel tool is a symmetry-based 29Si dipolar recoupling method (SR264n) [123] for small dipolar interactions that has been initially applied in zeolite structural studies by Brouwer et al. [124], One of the advantages of the new method over INADEQUATE is that the latter misses auto-correlations of symmetry-related double-quantum coherences. The SR26411 method provides such information on auto-correlation which allows to identification of all four connectivities of a tetrahedral Si position. 

DP-2) These pulses are chosen in such a way that their auto-correlations are close to a spike with small side-lobes ... 

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

The double commutator [[g, Tr /) (/], Tlp q may form new operators different from Q, and some of these new operators may not even be physical observables. When the double commutator conserves the operator Q, one speaks of the auto-correlation mechanism. Otherwise, one speaks of the cross-relaxation process. In other words, cross-relaxation is independent of the nature of the relaxation mechanism, but involves the interconversion between different operators. To facilitate such a possibility, it is desirable to write the density operator in terms of a complete set of orthogonal basis... 

In homogeneous isotropic turbulence, the two-point velocity correlation function can be expressed (Pope 2000) in terms of the longitudinal (/) and transverse (g) auto-correlation functions ... 

R,j (r. t) is completely determined by the longitudinal auto-correlation function f(r, t). 

The auto-correlation functions can be used to define two characteristic length scales of an isotropic turbulent flow. The longitudinal integral scale is defined by... 

Isotropic turbulence is described by a single-time auto-correlation function pu(r). Thus, an integral time scale can be defined in terms of the auto-correlation function by... 

The latter is difficult to discern from file figure, but appears clearly in file Lagrangian auto-correlation functions shown in Yeung (2002). 

Alternatively, the white-noise processes W(f) could be replaced by colored-noise processes. Since the latter have finite auto-correlation times, the resulting Lagrangian correlation functions for U and would be nonexponential. However, it would generally not be possible to describe the Lagrangian PDF by a Fokker-Planck equation. Thus, in order to simplify the comparison with Eulerian PDF methods, we will use white-noise processes throughout this section. 

The Lagrangian auto-correlation time T0, found from (6.268) is inversely proportional to Cmi. The latter can be fit to the DNS data of Yeung (2001) as ... 

Note that Tm decreases with increasing Reynolds number. As discussed in Fox and Yeung (2003), the auto-correlation time of co has a direct effect on the model predictions for the scalar dissipation rate (described below). Thus, in applications involving scalar mixing, it is best to use the Reynolds-number-independent value of Cml = 5, originally suggested by Dreeben and Pope (1997b). 

In this section, we will briefly describe the two main techniques devoted to detecting ultrashort pulses the streak camera and the auto correlator. 

The auto correlator is an important tool in the detection of nltrashort pulses in the femtosecond-picosecond range. The basic operational scheme of an autocorrelator is shown in Figure 3.28. 

Fig. 1. Auto-correlation function of the energy and the best fit for a double exponential decay used to obtain the interval of statistical correlation.

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