Entropy autocorrelation

Formally, S2 represents a decrease in the autocorrelation function caused by the motion S2=0 corresponds to completely unrestricted motion of a bond (N-H in this case), while S2=0 is expected if the bond reorientations are frozen. It was shown recently that the order parameter may be related to the statistical mechanical properties of a protein molecule [33-35] hence, changes in the NMR-derived order parameters can indicate localized contributions to overall molecular entropy. 

The concept behind the maximum entropy model is to choose the spectrum in the form of a non-negative function of frequency, which corresponds to a time series with maximum entropy whose autocorrelation function is consistent with the set of known values. 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

Interplay of attractive surface contacts and loss of configurational entropy 25 suggest that the properties of polymers at surfaces can be different from those in bulk. The analysis of the intensity autocorrelation requires evaluation of Eq. (10) without the assumptions that particles are independent and that averaging over... 

As mentioned earlier, it is difficult to obtain a quantitative measure of entropy. By using the 2PT method (the method will be described later in Chapter 19), one can obtain the entropy of water molecules in both major and minor grooves of DNA. One can also get a measure of the translational diffijsivity of those water molecules from the mean-square displacement or velocity autocorrelation function - all these are fortunately easily available with computer simulations. 

The analysis involved deconvolution by iterative reconvolution, background subtraction, and optional correction for shift of the instrument response function. Statistical tests included chi-square, the Durbin-Watson test, the covariance matrix, a runs test, and the autocorrelation function [6]. An alternative form of data analysis involves distributions of lifetimes rather than a series of exponentials. Differentiation of systems obeying a decay law made up of three discrete components from systems where there exists a continuous distribution of lifetimes, or a distribution plus one or more discrete components, is a nontrivial analytical problem. Methods involving the minimization of the chi-square parameter are commonly used, but recently the maximum entropy method (MEM) has gained popularity [7]. Inherent in the MEM method is the theoretical lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Recent work has compared the MEM with a new version of the exponential series method (ESM) which allows use of the same size probe function as the MEM and found that the two methods gave comparable results [8]. 

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