Auto-correlation functions

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... 

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). 

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.

Even without having the stmcture factor phases, e.g. from electron microscopy images, it is possible to get some insight into the atomic architecture of a crystal. A simple but powerful method to get this information was introduced hy A.L. Patterson about 70 years ago. Following Patterson the Fourier synthesis is carried out using the squared stmcture factor amplitudes Fha which are equal to the measured intensities for the reflections with index hkl. Moreover, all phase values must be set to zero, which leads to the following (auto-correlation) function ... 

D. Axelson, Florida State University, Florida One comment and then one question. The polyethylene sample you showed in the last slide I ve taken below i 5°sample develops a broad distribution or non-exponential auto correlation function at low temperatures. One of the giveaways is that in going through the Ti minimum if it is not 110 ms at 67 MHz and if a... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Physically this description corresponds to putting an atom (mass M) in an external time-dependent harmonic potential (frequency co0). The potential relaxes exponentially in time (time constant l/x0) so that eventually the atom experiences only a frictional force. Compared with other models2 which have been proposed for neutron scattering calculation, the present model treats oscillatory and diffusive motions of an atom in terms of a single equation. Both types of motion are governed by the shape of the potential and the manner in which it decays. The model yields the same velocity auto-correlation function v /(r) as that obtained by Berne, Boon, and Rice2 using the memory function approach. 

Note that in the above expression the memory function is proportional to the auto-correlation function of the random force. This is the well-known second fluctuation-dissipation theorem. 

A convenient and accurate analysis of these intensity fluctuations is performed by computing the auto-correlation function of the measured intensities, G(x), where... 

Fig. 2. PD results input and output auto correlation functions with Vci = 40m3.

A better correlated input signal will leave less space for improvement compared to the previous. Fig. 3 shows the improvement of the auto correlation functions for Vc = 400 m3. It follows that the characteristic volume of the output property (Fco) is just slightly higher compared to Vci = 40 m3 Vco 3400 m3 versus Vco 3200 m3 for a = 0° and /S = 30°. 

The slightly underestimation of the homogenization efficiency in Table 1 and the slightly overestimation in Table 2 can be a result of the realization of the stochastic input properties, although these are based on the analytical input auto correlation function used for PD. Another possible cause is the number of layers in practice the layer thickness is 10 cm, this value was used in the simulations. But, PD is based on infinite thin layers so using layers of 10 cm can make the difference with the volume distribution of different layers in a slice (Section 2). 

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