Correlation functions definition

The definition of the terms can be found in Refs. 91, 92. The excess chemical potential, computed from the direct correlation function of the... 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

Correlation functions measure the relationship between two variables, x and y. A common definition is... 

Using solution (1.37) in definition (1.4), one has the angular momentum correlation function... 

It is evident from this that the most likely terminal position is x(x, x) = —Q(i)Sx, as expected from the definition of the correlation function, and the fact that for a Gaussian probability means equal modes. This last point also ensures that the reduction condition is automatically satisfied, and that the maximum value of the second entropy is just the first entropy,... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

The result for the like and unlike partners definition can be obtained by very similar arguments and involves all three pair correlation functions. The various definitions and results can equally be applied to defects which are not ionic by merely substituting the words different kind for opposite charge and either kind for either charge in the definitions. 

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... 

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

As has been mentioned above, the inclusion of basis functions (49) with high power values, nik, is very essential for the calculations of molecular systems. It is especially important for highly vibrationally excited states where there are many highly localized peaks in the nuclear correlation function. To illustrate this point, we calculated this correlation function (it corresponds to the internuclear distance, r -p = r ), which is the same as the probability density of pseudoparticle 1. The definition of this quantity is as follows ... 

The limit of this expression gives us the familiar definition for the cross-correlation function with the limits of integration redefined for a distribution about zero. 

Some basic definitions are necessary. The definition of a cross-correlation function (CCF) of two non-zero average power signals, x(t) and y(t), is ... 

The definition of correlation functions in this book differs from the definition of the correlation coefficient in the theory of probability. The difference is essentially in the normalization, i.e., whereas g(, ) can be any positive number 0 S g the correlation coefficient varies within [-1,1]. We have chosen the definition of correlation as in Eq. (1.5.19) or (1.5.20) to conform with the definition used in the theory of liquids and solutions. 

At this point we again stress the sequence of definitions leading to Eq. (4.2.16). First, the correlation function is defined as a measure of the extent of the dependence between the two events in Eq. (4.2.12) [or, equivalently, in Eq. (4.2.13)]. The probabilities used in the definition of g a, b) were read from the GPF of the system, e.g., (4.2.1). This side of g a, b) allows us to investigate the molecular content of the correlation function, which is the central issue of this book. The other side of g a, b) follows from the recognition that the limiting value of g(a, b), denoted by g a, b), connects the binding constants ah and kg A. This side of g a, b) allows us to extract information on the cooperativity of the system from the experimental data. In other words, these relationships may be used to calculate the correlation fimction from experimental data, on the one hand, and to interpret these correlation functions in terms of molecular properties, on the other. 

An example that conforms to this definition would be benzene-l,3,5-tricar-boxylic acid (Section 5.9). Clearly, owing to the symmetry of the molecule there is only one intrinsic binding constant k, and only one intrinsic binding constant for pairs fcj, or, equivalently, only one pair correlation function... 

Another useful definition of GOT is the correlation function as defined by... 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

It follows from the definition of the functionals and Ti that the exchange-correlation functional is invariant under the rotation and translation of the electron density. This follows directly from the fact that the kinetic energy operator f and the two-electron operator W are invariant with respect to these transformations. This also has implications for the exchange-correlation functional. 

Using the definition < A > = Tr(pA), we can express all the matrix elements in the density matrix in terms of different spin-spin correlation functions [62] ... 

Here Q2 is the average value of the square of the oscillation amplitude, K(r) is the correlation function of the random process, and Vf are definite functions of the 4/electron coordinates (40). 

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