Distribution functions, Gaussian trial function

Data from phase-modulation fluorometry have been analyzed using an alternative approach to those described above, as expounded by Gratton and co-workers(14 12 13,22) and Lakowicz et al. W> Here, Lorentzian or Gaussian distribution functions with widths and centers determined by least-squares analysis are used to model the unknown distribution function. While this approach may introduce assumptions about the shape of the ultimate distribution function since these trial functions are symmetric, it has the advantage of minimizing the number of parameters involved in the fit. Here, a minimum x2 is sought, where... 

An important issue is to verify that the energy differences are normally distributed. Recall that if the moments of the energy difference are bounded, the central limit theorem implies that given enough samples, the distribution of the mean value will be Gaussian. Careful attention to the trial function to ensure that the local energies are well behaved may be needed. 

In addition to approximations for c(1,2 pl), a suitable parametrization of the trial function n(r,m) is required. The position dependence can be described in full detail by a Fourier expansion, the form of which is determined by the choice of crystal symmetry for the solid phase. More simply, a Gaussian distribution of molecular density about the sites of the crystal lattice may be assumed the accuracy of this latter approximation has been verified for... 

An additional issue in the development of the density functional theory is the parameterization of the trial function for the one-body density. Early applications followed the Kirkwood-Monroe [17,18] idea of using a Fourier expansion [115-117,133]. More recent work has used a Gaussian distribution centered about each lattice site [122]. It is believed that the latter approach removes questions about the influence of truncating the Fourier expansion upon the DFT results, although departures from Gaussian shape in the one-body density can also be important as has been demonstrated in computer simulations [134,135]. 

Comparison of these results with exact values (6.12) and (6.14) shows that the trial function chosen by Onsager works quite well. Odijk [15, 16] realized that for large values of k, Onsager s orientational distribution function can be approximated by a Gaussian distribution function... 

The featureless appearance of flic experimental spectra results from a broad statistical distribution of the parameters. In order to evaluate die eft parameters from such spectra, the numerical simulations should incorporate two distributions simultaneously, those of D and EID (die use of independent Gaussian distributions of D and E in [35] was, strictly speaking, incorrect because such an approach did not enforce the condition that EID < 1/3). The problem dien arises as to which mathematical functions can be used to describe die distributions ofD and EID, and are these distributions correlated or not. While these questions could possibly be dealt with using the method of trial and error, die resulting formal solutions would still require a physical justification. Therefore, we have approached the problem from the opposite direction, by first considering a physical model dial would relate... 

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