Autocorrelation, homogeneous

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

This equation cannot be solved analytically due to the volume distribution. Therefore we have built a simulation program to calculate the homogenization efficiency (ay/ax and the improvement of the autocorrelation function). 

This study has shown the possibility to measure segregation problems when discharging mixtures through a funnel. It has also shown the effect of the number of drum revolutions at a fixed speed on the quality of a mixture, as well as the effect of a static mixer. The axial structure of the mixtures through autocorrelation functions could also be studied from these data, but this has not been reported here for clarity of the paper. In addition, it must be remembered that we just studied the evolution of the dielectric permittivity, and that in most cases, we will have to follow the volumetric compositions of each component in order to characterise the homogeneity of the medium. The capacitive method is indeed full of promise for particulate systems, and it would be interesting to explore it much into details, particularly for determining the proportions of each component of the mixture. 

Since the particle motion is time homogeneous (i.e., the autocorrelation function for the energy of the particle is independent of the initial time), the above equation can be rearranged in the form... 

Thus, for homogeneous and isotropic systems, g(r) is the autocorrelation function of the local particle-concentration. 

E>vib and Our also show up in the theory of spontaneous Raman spectroscopy describing fluctuations of the molecular system. The functions enter the CARS interaction involving vibrational excitation with subsequent dissipation as a consequence of the dissipation-fluctuation theorem and further approximations (21). Equations (2)-(5) refer to a simplified picture a collective, delocalized character of the vibrational mode is not included in the theoretical treatment. It is also assumed that vibrational and reori-entational relaxation are statistically independent. On the other hand, any specific assumption as to the time evolution of vib (or or), e.g., if exponential or nonexponential, is made unnecessary by the present approach. Homogeneous or inhomogeneous dephasing are included as special cases. It is the primary goal of time-domain CARS to determine the autocorrelation functions directly from experimental data. 

For pulses much longer than T2, the expression for the scattered energy is simply the envelope of the electric field autocorrelation squared for both the homogeneous and inhomogeneous cases ... 

The S2 absorption spectrum is displayed in Fig. 15. Panel (a) compares the semiclassical and the quantum result and panel (b) shows the experimental data from Ref. 163. Both theoretical results have been obtained by Fourier transformation of the autocorrelation function. Thereby a phenomenological dephasing constant T2 = 30 fs has been included to reproduce the homogeneous width of the experimental spectrum. The absorption spectrum shows a diffuse S2 band with irregularly spaced structures, which cannot be assigned in terms of normal modes in the S2 state. The... 

Let us come back now to the question of increasing heterogeneity in the local mobilities upon decreasing the temperature. We have already identified a tendency for immediate back jumps after one torsional transition as the reason for the different temperature dependencies of the mean waiting time between torsional transitions (twait) and the torsional autocorrelation time ttacf)- In a homogeneous system, where every chemically identical torsion shows identical dynamics on the time scales of observation the probability distribution of waiting times should be... 

Examples of various correlation functions for the diblock copolymer system at high and at low temperatures are shown in Fig. 7. It has been observed that at high temperatures TjN = I), the systems behave like a homogeneous melt. All correlation functions show a single step relaxation. The fastest is the bond relaxation and the slowest is the chain relaxation described by the end-to-end vector autocorrelation function. The relaxation of the block is faster than the whole chain relaxation by a factor of approximately two. Such relations between various relaxation times in the disordered state of the copolymer can be regarded as confirmed experimentally for some real systems, in which the dielectric spectroscopy allows distinction of the different relaxation modes [41]. At low temperatures, drastic changes... 

The example is a 32-feet long, fixed ended beam with stochastic flexural rigidity, EI(x which is subject to a distributed load with stochastic intensity W(jc), as shown in Fig. 1. Both processes are assumed to be homogeneous and Gaussian with the means and coefficients of variation listed in Table 1. The autocorrelation coefficient functions are assumed to be of the form... 

By extrapolation of the computed autocorrelation coefficient to zero time the square of the llagrangian fluctuating velocity can be obtained. The square root of this quantity can then be compared to the Eulerian rms velocity for the same column compartment. The two velocities should be equal in case of homogeneous turbulence. Devanathan (1991) shows that this is not the case in bubble columns but that homogeneous turbulence is approached at the highest gas velocities in the largest diameter column. 

For systems in homogeneous environments and in the absence of cross-correlations between different chains, crr 1) is reduced to the autocorrelation function of the end-to-end vector. 

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