Gaussian distribution information approach

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

The higher even moments of the unperturbed dimensions, r P)Q and s p)q, p > 1, are accessible through an appropriate expansion in the dimensions of the generator matrices used for the simpler cases where p = 1 [11]. Dimensionless ratios formed from appropriate combinations of these even moments provide information about the shape of the distribution functions. Thus r ) measures the average value of r, r )dl(r )o measures the width of the distributirMi for and (r )o/ r )l measures the skewness of this distribution function. All flexible homopolymers wiU approach the Gaussian limit of = 5/3 as n 00, but nar-... 

Calculations of model spectra (distribution functions/(6jj) of chemical shifts of protons) using Gaussian functions and parameters of dispersion of peaks from the experimental NMR spectra (or theoretical estimations) as described in Chapter 10. Such approach allows us to calculate appropriate NMR spectra of large systems. This information can be used for more reliable and detailed analysis of the experimental NMR spectra. 

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