Lack of self-averaging

Milchev A, Binder K and Heermann D W 1986 Fluctuations and lack of self-averaging in the kinetics of domain growth Z. Phys. B. Condens. Matter. 63 521 -35... 

However, when we consider a problem involving a single flexible polymer chain with N effective monomers and take the thermodynamic limit, N—> =, we find a lack of self-averaging when we consider quantities like the mean square end-to-end distance (R ) (for a free chain in solution, = 0 because all direaions for the orientation of the end-to-end veaor are equivalent, of course). For the case of Gaussian chains, we have a probability distribution (in d dimensions)... 

Note that this result does not decrease at all when we increase N the only way to decrease the error is to increase the number M of statistically independent samples. For chains in the presence of excluded volume, Pn(R) no longer is a Gaussian, but lack of self-averaging prevails only the constant 2/d in eqn [26] is replaced by a different number. 

Gaxel, T. and Monthus, C. (2005b). Distribution of Pseudo-critical Temperatures and Lack of Self-averaging in Disordered Poland-Scheraga Models with Different Loop Exponents, Eur. Phys. J. B 48, pp. 393-403. 

The improvement came in the form of the coherent-potential approximation (CPA) (Soven 1967, Taylor 1967, Velicky et al 1968), which remedied the lack of self-consistency exhibited by the ATA. The crux of this approach is that each lattice site has associated with it a complex self-consistent potential, called a coherent potential (CP). The CP gives rise to an effective medium with the important property that removing that part of the medium belonging to a particular site, and replacing it by the true potential, produces, on average, no further scattering. Because the CPA is used for our discussion of chemisorption on DBA s, its mathematical formulation is given below. 

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The site-site RISM/HNC theory has been coupled with the ab initio molecular orbital (MO) theory in a self-consistent field (SCF) calculation of the electronic and solvation structure of a solute molecule immersed in molecular solvent, referred to as the RISM-SCF method [59, 60, 61]. Since the site-site treatment of the solute-solvent correlations involves the approximation of radial averaging, it constitutes a bottleneck of the RISM-SCF method. Although this approach yields reasonable results for the thermodynamics of solvation for many solute species and solvents [62], it lacks a 3D picture of the solvation structure for complex solutes and oversimplifies the contribution to the solvation properties from highly directed electron orbitals of the solute molecule. 

Bulk polymer properties such as viscosity and elasticity are concerned with averaged responses of an assembly of polymer chains to external stimuli. On the other hand, the self-diffusion coefficient has something to do with the average speed of translation of the centers-of-mass of individual chains. Thus its study should give us a clue to the clarification of the modes of Brownian motion of a single chain on long timescales. This expectation must have been in the mind of polymer workers for many years, but, except in dilute solutions, few measurements of Ds were undertaken until recently, probably on the one hand because of experimental difficulties and on the other because of the lack of an adequate guiding theory. 

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