Power-exponential correlation function

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. 

Figure 6.2 Plot of exponential, Gaussian, and power serial correlation functions as a function of distance between measurements. The underlying correlation between measurements was 0.8.

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... 

The time dependence of the dynamic correlation function q t) was investigated numerically on the Ising EA model by Ogielski [131], An empirical formula for the decay of q t) was proposed as a combination of a power law at short times and a stretched exponential at long times... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Figure 5. Exact C(f, t ) for case 2 exponential on times and power-law off times with a = 0.4. We use 4r+(i) = 1/(1 + s) and fr (j) = 1/(1 + s -4) and numerically obtain the correlation function. For each curve in the figure we fix the time t. The process starts in the state on. Thick dashed straight line shows the asymptotic behavior Eq. (21). For short times (t < 1 for our example) we observe the behavior C(tff) C(f, 0) = (/(f)), the correlation function is flat.

When the correlation function of inverse power law, the relaxation of the pointer can be evaluated with the trajectory method, and it is proven to be an exponential decay followed by oscillations whose intensity decays with an inverse power law [59]. 

With local information given by INM analysis in mind, we next see the character of rotational relaxation in liquid water. The most familiar way to see this, not only for numerical simulations [76-78] but for laboratory experiments, is to measure dielectric relaxation, by means of which total or individual dipole moments can be probed [79,80]. Figure 10 gives power spectra of the total dipole moment fluctuation of liquid water, together with the case of water cluster, (H20)io8- The spectral profile for liquid water is nearly fitted to the Lorentzian, which is consistent with a direct calculation of the correlation function of rotational motions. The exponential decaying behavior of dielectric relaxation was actually verified in laboratory experiments [79,80]. On the other hand, the profile for water cluster deviates from the Lorentzian function. As stated in Section III, the dynamics of finite systems may be more difficult to be understood. 

It seems natural that we regard Cp(t x) as a series of stretched exponential functions of t rather than power-type functions, since this function fits Cp(t x) in more than two decades of time (power-law fits of the correlation functions hold in one decade). Moreover, at equilibrium, Cp(t teq) is also a stretched exponential rather than a pure exponential, as shown in Fig. 9. 

The atom-multiphonon component of the inelastic scattering can be obtained from the theory of Manson [44] by subtracting off the zero- and first-order terms from the power series expansion of the exponential displacement correlation function. With the assumption that the major contribution to the multiphonon scattering is due to low-energy, long-wavelength phonons, he is able to arrive at an expression with a very simple form for the transition rate. 

When there are more than one species present in a sample, each contributes to give a correlation function which is a sum of exponentials and a power spectrum which is a sum of Lorentzians. The intensity of each component in the composite function is proportional to the product of the molecular weight and the concentration in (w/v) units, assuming the refractive index increments of each component are identical. Two basic approaches are available to extract the particle distribution from the QLLS data. 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

Figure 1.20 (A) Site-specific R relaxation rates of sRii from Anabaena sp. PCC7120 (ASR) determined at 12 kHz spinlock power and (B) motional correlation times estimated using single exponential autocorrelation function approximation. Repr/nfed from [250], Copyright 2014 American Chemical Society.

An important property of the self-avoiding chain is that the segment orientational correlation function decays at large distances not exponentially, like in ideal chain (eqn [7]), but as a power-law function ... 

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