Spatial correlation function

Numerous methods for characterising the earliest stages of phase separation have been devised. All rely on sampling the data to make estimates of the local composition. The aim is to be able to identify statistically significant fluctuations in solute concentration which indicate the onset of phase separation. A comprehensive review of the key methods, composition frequency distribution, contingency table analyses, pan-correlation functions, spatial distribution mapping and the local chemistry approach is presented in Marquis and Hyde. Pair correlations. 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

The two-point spatial correlation function C2(r) (equation 3.26) does not decay exponentially with distance r, but appears instead to decay more slowly a.s exp(—Q- /r). 

TOWARDS THE HYDRODYNAMIC LIMIT STRUCTURE FACTORS AND SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms of the spatial Fourier ( ) components of the density, particle currents, stress, and energy fluxes. The time correlation function of those Fourier components detail the decay of density, current, and fluctuation on the length scale of the Ijk. 

Several structure sizes caused by microphase separation occurring in the induction period as well as by crystallization were determined as a function of annealing time in order to determine how crystallization proceeds [9,18]. The characteristic wavelength A = 27r/Qm was obtained from the peak positions Qm of SAXS while the average size of the dense domains, probably having a liquid crystalline nematic structure as will be explained later, was estimated as follows. The dense domain size >i for the early stage of SD was calculated from the spatial density correlation function y(r) defined by Debye and Buche[50]... 

It is thus entirely expressed in terms of the zero wave number Fourier coefficient pQ(p t). Similarly, the pair correlation function in a spatially homogeneous system is defined by17... 

The spatial correlation functions are computed from the two-point joint velocity PDF based on two points in space. Obviously, the same idea can be extended to cover two points in time. Indeed, the Eulerian9 two-time joint velocity PDF /Lf (J (V, V x, t, t ),... 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

This relation shows that for homogeneous turbulence, working in terms of the two-point spatial correlation function or in terms of the velocity spectrum tensor is entirely equivalent. In the turbulence literature, models formulated in terms of the velocity spectrum tensor are referred to as spectral models (for further details, see McComb (1990) or Lesieur (1997)). 

The Fourier transform introduces the wavenumber vector , which has units of 1 /length. Note that, from its definition, the velocity spatial correlation function is related to the Reynolds stresses by... 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

Similar Fourier transform pairs relate the spatial correlation functions defined in (3.40) and (3.41) to corresponding cospectra t) and t), respectively. 

Note that from its definition, the scalar spatial correlation function is related to the scalar variance by... 

The photoinduced absorbance anisotropy in a TPD experiment relaxes according to the same correlation function as in Eq. (4.16).(29) Effects of spatial variations in the excitation and probe beams, and chromophore concentration, have been treated and shown not to alter the final result.(29) NMR dipolar relaxation rates are expressed in terms of Fourier transforms of the correlation functions, 4ji< T2m[fi(0)] T2m[i2(f)]>> where fl(f) denotes the orientation of a particular internuclear vector. In view of Eq. (4.7), these correlation functions are independent of the index m, hence formally the same as in Eq. (4.16). For the analysis of NMR relaxation data, it is necessary also to evaluate Fourier transforms of the correlation functions. Methods to accomplish this in the case of deformable DNAs have been developed and applied to analyze a variety of data.(81 83)... 

Chapter 4 deals with the local dynamics of polymer melts and the glass transition. NSE results on the self- and the pair correlation function relating to the primary and secondary relaxation will be discussed. We will show that the macroscopic flow manifests itself on the nearest neighbour scale and relate the secondary relaxations to intrachain dynamics. The question of the spatial heterogeneity of the a-process will be another important issue. NSE observations demonstrate a subhnear diffusion regime underlying the atomic motions during the structural a-relaxation. 

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 plume structure data introduced in the previous sections provide the opportunity to assess whether multiple sensors that are spatially separated can rapidly acquire useful information [5], The correlation function for the instantaneous concentration acquired by two sensors at locations p and q, separated in the transverse direction, is defined as... 

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