Exponential correlation function

A possible step in this direction can be made through use of earlier relaxation studies on other systems. Hunt and Powles,— when studying the proton relaxation in liquids and glasses, found the relaxation best described by a "defect-diffusion" model, in which a non-exponential correlation function corresponding to diffusion is Included together with the usual exponential function corresponding to rotational motion. The correlation function is taken as the product of the two independent reorientation pro-... 

The non-Markovian heat bath presents an exponential correlation function characterized by rc. 

In order to compare the results from PD with the simulations, the input properties, and silo geometry have to be the same. As the results of PD in the previous section are based on a first order negative exponential correlation function, this function was used as well for the simulations. Three different input properties were used with different characteristic volumes (Vci) 40 and 400 m3. The silo volume was approximately 40,000 m3, with a height of 50 m and a diameter of 32 m. This relatively high silo was chosen because constant angles over the silo height are required (as indicated in Fig. 1) for a thorough comparison with PD. As the input properties are realizations of a stochastic process 400 repetitions were done per simulation. 

In this Table we also find a second form of exponentially correlated function s]. Such functions have been referred to as explicitly correlated Gaussians (ECG). They take the form,... 

The correlation function R(-, ) in (2) is central to this statistical model. The power-exponential class of correlation functions is a popular choice, for its computational simplicity and because it has been successful in many applications. The power-exponential correlation function is... 

The power-exponential correlation function (3), for example, is of this product form. To computere x), the integral on the right-hand side of (22) is evaluated as... 

In this chapter, we use a type of initial condition that is different from the waterbag used in Refs. 15 and 18, and we show that (i) probability distribution functions do not have power-law tails in quasi-stationary states and (ii) the diffusion becomes anomalous if and only if the state is neither stationary nor quasi-stationary. In other words, the diffusion is shown to be normal in quasi-stationary states, although a stretched exponential correlation function is present instead of usual exponential correlation. Some scaling laws concerned with degrees of freedom are also exhibited, and the simple scaling laws imply that the results mentioned above holds irrespective of degrees of freedom. 

By using the fitting function (16) and Eq. (15), we numerically reproduce Gg(f feq), and the reproduced curve well approximates the numerical result as shown in Fig. 9b. Note that cjg(f feq) is proportional to f2 in the limit of f 0, since Cp(s feq) in Eq. (15) goes to the constant Cp(0 feq). On the other hand, in the limit of f —> oo, cjg(f feq) is proportional to f, because both Cp(s feq) and sCp(s teq) are almost zeros in the long-time region, and hence their integrals become constants. The crossover from f2 to f is also observed if we assume an exponential correlation function, and hence we conclude that diffusion at equilibrium is normal as expected although a stretched exponential is present. 

Using the approximate functions (19) and Eq. (12), we are able to reproduce cig(t), as shown in Fig. 12. The approximation is good in Stage I—that is, in the quasi-stationary time region—irrespective of the value of N. Consequently, there is no anomaly in diffusion in Stage I, since the diffusion is explained by stretched exponential correlation function. 

In many cases the auto-correlation function is an exponential function with a time constant Tc, which is called the correlation time of the process. Following the definition of the correlation time for an exponential correlation function (3.2.10) the correlation time for a nonexponential correlation function is defined as... 

In the short-time limit and with an exponential correlation function the FID s(t) is approximated by... 

Assuming an exponential correlation function exp(- t /T ) with correlation time depending on the phase, we finally obtain... 

For a simple exponential correlation function, given in Eq. [18], the corresponding spectral density is a Lorentzian... 

Often, the most that is known about a bath is a relevant relaxation time. In that case, the use of a simple exponential correlation function might be appropriate. If one makes the further assumption that the autocorrelation functions of the random variables decay rapidly compared to all other important time scales in the system, simplifies to [38,40]... 

The total transverse relaxation function M(t) for inter-crosslinked chains and dangling chain ends follows an exponential correlation function [58], as shown in Equation 9.10 ... 

In the transition region, highly non-exponential correlation functions were observed but it was not possible to resolve these data into individual components ". ... 

Bases of fully exponentially correlated wavefunctions [1, 2] provide more rapid convergence as a function of expansion length than any other type of basis thus far employed for quantum mechanical computations on Coulomb systems consisting of four particles or less. This feature makes it attractive to use such bases to construct ultra-compact expansions which exhibit reasonable accuracy while maintaining a practical capability to visualize the salient features of the wavefunction. For this purpose, exponentially correlated functions have advantages over related expansions of Hylleraas type [3], in which the individual-term explicit correlation is limited to pre-exponential powers of various interparticle distances (genetically denoted r,j). The general features of the exponentially correlated expansions are well illustrated for three-body systems by our work on He and its isoelectronic ions, for... 

EOM-CC method (p. 638) exchange hole (p. 597) explicit correlation (p. 584) exponentially correlated function (p. 594) Fermi hole (p. 597) frozen orbitals (p. 624)... 

The relaxation dynamics of junctions in polymer networks have not been well-known until the advent of solid-state NMR spin-lattice relaxation measurements in a series of poly(tetrahydrofuran) networks with tris(4-isocyanatophenyl)-thiophosphate junctions [100]. The junction relaxation properties were studied in networks with molecular weights between crosslinks. Me, ranging from 250 to 2900. The dominant mechanism for nuclear spin lattice relaxation times measured over a wide range of temperatures were fit satisfactorily by spectral density functions, /( ), derived from the Fourier transforms of the Kohlrausch stretched exponential correlation functions... 

The Guinier, Debye-Bueche, Invariant and Porod analyses are all based on the assumption of well defined phases with sharp interfacial boundaries. In addition, the Guinier approach is based on the assumption that the length distribution function (23.15), or probability Poo(r) that a randomly placed rod (length, r) can have both ends in the same scattering particle (phase) is zero beyond a well defined limit. For example, for monodisperse spheres, diameter D, Poo = 0, for r > D. In the Debye-Bueche model, Poo has no cut off and approaches zero via an exponential correlation function only in the limit r oo [45,46]. 

In this approximation, the correlations of the order parameter fluctuations are described with the exponential correlation function within long-range correlations f. I he level of the order parameter fluctuations allowing the mean field approximation is determined by the Ginzburg parameter. 

A numte of theoretical models have been derived specifically for local chain motions [82]. These models have been usehil in stimulating thinking about the underlying mechanisms for local dynamics, and about how these mechanisms influence the shape of the correlation function. For example, it is now understood that non-exponential correlation functions can result from a variety of sources, including specific cooperative motions. Any model which resembles a one-dimensional diffusion equation will have a non-exponential decay [81]. 

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