Correlation function decay

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

Lindenberg and West conclude, after analysis of Eq.(59) at low temperatures where kTccIly, that the correlation function decays on a time scale li/ kT rather than 1/y. Thus, the bath can dissipate excitations whose energies lie in the range (0/fi.y), while the spontaneous fluctuations occur only in the range (0,kT) if kTcorrelation time of the fluctuations is therefore the longer of fi/ kT and 1/y. The idea advanced by these authors is that fluctuations and dissipation can have quite distinct time scales [133], This is important if the two quantum states of the system of interest correspond to chemical interconverting states [139, 144, 145],... 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

It is the spread of oscillation frequencies a>j that causes the environment response to decohere after a (typically short) correlation time t (Figure 4.3b). Hence, the Markovian assumption that the correlation function decays to 0 instantaneously, d>(t) 8 t), is widely used it is, in particular, the basis for the... 

The movements capable of relaxing the nuclear spin that are of interest here are related to the presence of unpaired electrons, as has been discussed in Section 3.1. They are electron spin relaxation, molecular rotation, and chemical exchange. These correlation times are indicated as rs (electronic relaxation correlation time), xr (rotational correlation time), and xm (exchange correlation time). All of them can modulate the dipolar coupling energy and therefore can cause nuclear relaxation. Each of them contributes to the decay of the correlation function. If these movements are independent of one another, then the correlation function decays according to the product... 

The direct correlation function decays fairly rapidly with the distance. However, this information, which correlates the structure to the interactions in the fluid, is too poor to provide a complete and accurate theory. 

The correlation function decays exponentially as r increases, differing from the one-dimensional model in that the pre-exponential factor also depends on r. 

Our results suggest that the spin correlation functions decay exponentially with a correlation length 1 for an arbitrary parameter a. We also assume that the decay of the correlation function is of the exponential type for the 14 parameter model as well, i.e., for any choice of site spinor I>A/u/p. This assumption is supported in special cases 1) the partition of the system into one-dimensional chains with exactly known exponentially decaying correlation functions 2) the two-dimensional AKLT model, for which the exponential character of the decay of the correlation function has been rigorously proved [32], Further evidence of the stated assumption lies in the numerical results obtained for various values of the parameter in the one-parameter model. 

If we have one condition with correlation function decaying to another with correlation function the resultant will, in general, be a convolution. For the present situation, however, y arises from osdllation in a small range of an es while y involves complete reversals. Shimizu > has observed that in such a case... 

Unie resolved ground state hole spectra of cresyl violet in acetonitrile, methanol, and ethanol at room temperature have been measured in subpicosecond to picosecond time region. The time correlation function of the solvent relaxation expressed by the hole width was obtained. The main part of the correlation function decayed much slower compared with that of the reported correlation function observed in time dependent fluorescence Stokes shift. Some possible mechanisms are proposed for understanding of the time depencences of the spectral broadening under the condition with the distribution of the relaxation times in fluid solution based on the entropy term in the solvent orientation as well as the site dependent response of the solvent. 

Fig. 7 shows the time correlation functions of solvent fluctuation in accordance with eq (7) using the values plotted in Fig. 6. In this calculation we used a(0)=30cm as the spectral width of the exciting laser pulse and 1430cm 1360cm and 1250cm for a(o>) in acetonitrile, methanol, and ethanol, respectively. In Fig. 7 the reported correlation functions obtained by the dynamic Stokes shift measurements of LDS7S0 in acetonitrile and 1-aminonaphthalene in methanol in accordance with eq (6) are also plotted by dashed lines. According to the literatures -, the correlation function decayed more than 80% in acetonitrile in the first 200fs and about 60% in methanol in the first 500fe. The peak shift of the absorption spectra in the present work was completed in the first Ips in methanol and ethanol solutions as indicated in Fig. 5 (2). However, the peak shift in acetonitrile solution could not be observed in the time resolution of our system. It is obvious fiom above results that the major part of the energy relaxation due to the fast response of the solvent dynamics is taken place in a few...

For Li, the time correlation function decays slowly compared to Na", due to the strong Li -water interaction. Thus, at low temperatures we could not detect differences in the dynamics given by the 12-6 or 10-3 functions. However, at high temperature, the 10-3 potential yields shorter times than the 12-6. For Mg", we observe that the 6-4 function yields a larger coordination... 

Coming back to the timescale issue, it is clear that direct observation of signals such as shown in Fig. 13.2 cannot be achieved with numerical simulations. Fortunately an alternative approach is suggested by Eq. (13.26), which provides a way to compute the vibrational relaxation rate directly. This calculation involves the autocorrelation function of the force exerted by the solvent atoms on the frozen oscillator coordinate. Because such correlation functions decay to zero relatively fast (on timescales in the range of pico to nano seconds depending on temperature), its numerical evaluation requires much shorter simulations. Several points should be noted ... 

In this expression, P2 is the second Legendre polynomial and i(t) is a unit vector with the same orientation as the transition dipole at time t. The brackets indicate an ensemble average over all transition dipoles in the sample. The correlation function has a value of one at very short times when the orientation of y(t) has not changed from its initial orientation. At long times, the correlation function decays to zero because all memory of the initial orientation is lost. At intermediate times, the shape of the correlation function provides detailed information about the types of motions taking place. Table I shows the three theoretical models for the correlation function which we have compared with our experimental results. 

In a previous publication 17, we compared the experimental anisotropies for dilute solutions of labeled polyisoprene in hexane and cyclohexane to several theoretical models. These results are shown in Table II. The major conclusions of the previous study are 1) The theoretical models proposed by Hall and Helfand, and by Bendler and Yaris provide good fits to the experimentally measured correlation function for both hexane and cyclohexane. The model suggested by Viovy, et al. does not fit as well as the other two models. 2) Within experimental error, the shape of the correlation function is the same in the two solvents (i.e, the ratio of t2/ti is constant). 3) The time scale of the correlation function decay scales roughly with the solvent viscosity. 

For charged particles, the interactions are also long-ranged, but still there exists some Rcor beyond which no direct influence of the interaction is noticeable. There are several studies of the manner that the pair correlation function decays to unity [see for example Fisher and Widom (1969), Perry and Throup (1972)]. In any case, even when g(R) is of relatively long-range, we can still find a radius Rc beyond which the pair correlation is practically unity. 

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

In the solid phase, the Debye-Waller correlation function decays algebraically to zero for large r,... 

Above Tj, the sixfold bond orientational correlation function decays exponentially. 

While for q = 2 this is still identical to the Ising model and for q = 3 identical to the standard Potts model, a different behavior results for q > 4. In particular, the exponents for q = 4 can be non-universal for variants of this model (Knops, 1980). For q > 4 there occurs a Kosterlitz-Thouless phase transition to a floating phase at higher temperatures, where the correlation function decays algebraically, and a IC-transition to a commensurate phase with ordered structure at lower temperatures (Elitzur et al., 1979 Jose et al., 1977). 

That means for large t the correlation function decays towards the static average of the quantity Z. For systems exhibiting simple dynamics the decay from (Z2) to (Z)2 can be described by a single exponential law of the form... 

Moreover, when the probability flux correlation function decays on a time scale shorter than the time scale of the solvent effects on the R motion, the solvent effects on the R motion can be neglected. In this case, Cf,(t) can be approximated by the standard analytical expression for the time correlation function of an undamped quantum mechanical harmonic oscillator [53] ... 

These correlation functions decay to zero as the molecular orientation becomes randomized with respect to its initial value ... 

Watanabe and Klein have reported MD simulations of the hexagonal mesophase of sodium octanoate in water with hexagonal symmetry. The singlet (i.e., one atom) probability distribution functions of the carbon atoms on the hydrocarbon chains show close similarity to those in the micelle. The dynamics of water molecules close to the head groups shows lower mean square displacements, and their orientational correlation function decays more slowly than those of waters farther from the head groups, as was seen in a recent bilayer simulation.6 ... 

Fig. 2.2.2. The time-correlation function, . Initially this function is < 2>. For times very long compared to the correlation time, ta, the correlation function decays to 2. [cf. Eq.(2.2.5)].

This definition of /tc/ is not always useful. There are cases in which, for instance, the correlation function decays in such a way that /rc/ as defined by Eq. (2.2.11) is zero. For example, the correlation function may have positive and negative regions that cancel. In such cases Eq. (2.2.11) does not provide an adequate measure of the decay time. 

The velocity-correlation function decays to zero such that the integral has the asymptotic long-time form... 

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