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... 

The problems with eomputer generation of random surfaees are mostly eonneeted with small sizes of systems we ean use. Eaeh sueh a small system has somewhat different realization of the spatial distribution of heterogeneities. Only at the thermodynamie limit of a truly maeroseopie system does the self-averaging of extensive thermodynamie quantities oeeur. It means that, when using small systems, it is neeessary to generate several (often of... 

In the early days of alloy theory, Lifshitz argued on intuitive grounds that one can calculate the properties of a disordered alloy from one sufficiently large sample," and referred to this property of a large sample as self-averaging. It can be seen most easily in exact... 

In this paper, the electronic structure of disordered Cu-Zn alloys are studied by calculations on models with Cu and Zn atoms distributed randomly on the sites of fee and bcc lattices. Concentrations of 10%, 25%, 50%, 75%, and 90% are used. The lattice spacings are the same for all the bcc models, 5.5 Bohr radii, and for all the fee models, 6.9 Bohr radii. With these lattice constants, the atomic volumes of the atoms are essentially the same in the two different crystal structures. Most of the bcc models contain 432 atoms and the fee models contain 500 atoms. These clusters are periodically reproduced to fill all space. Some of these calculations have been described previously. The test that is used to demonstrate that these clusters are large enough to be self-averaging is to repeat selected calculations with models that have the same concentration but a completely different arrangement of Cu and Zn atoms. We found differences that are quite small, and will be specified below in the discussions of specific properties. 

This branch of polymer physics is closely connected to another of I.M. Lif-shitz favorite directions in physics, namely, the theory of disordered systems [73]. The situation when different samples have only statistical similarity is typical for the physics of chaos, and many concepts I.M. Lifshitz developed are also quite naturally applied in the physics of disordered polymers. The idea of self-averaging in general and self-averaging of free energy [74], in particular, are examples of such concepts. 

In the next section, we give some details on typical models that are studied, or problems that arise (correlations between subsequently generated configurations, finite size effects, self-averaging of observables, etc.) and how they are overcome (more details can be found in Refs. 2-4). 

Note that Eq. (25) implies that the square brackets occurring in Eq. (26) are of order l/N, off critical points since there x> C x converge to finite values independent of N. Thus densities of extensive quantities such as T, < )t> are self-averaging On the other hand, the response functions sampled from fluctuations, Eqs. (25), are not self-averaging their relative error is independent of system size e.g., for T>T where = 0, we have... 

The key point of these methods is the concept of self-averaging in accordance with this concept, the observable value of an additive function, for example - of the free energy, F, of macroscopic sample of spin glass, practically coincides with the averaged value over the ensemble of disorder realizations ... 

The phenomenon of self-averaging takes place for the systems with sufficiently small long-range correlations only in this case one can consider F as a sum of contributions from different volume elements, containing different and statistically independent realizations of disorder (for more details see Ref. [63]). 

Unlike that for the classical linear responses of such solids, the extreme nature of the breakdown statistics, nucleating from the weakest point of the sample, gives rise to a non-self-averaging property. We will discuss these distribution functions F a), or F(/), or F E) giving the cumulative probability of failure of a disordered sample of linear size L. We show that the generic form of the function F a) can be either the Weibull (1951) form... 

The means for improving the real space, electron density map may involve a combination of ideas, but by far the most powerful is application of noncrystallographic symmetry. If the asymmetric unit contains noncrystallographic symmetry, then its electron density map does as well. If the dispositions of the noncrystallographic symmetry operators are known, then the electron density map can be self-averaged using these operators. 

For many critical systems CLT may not be applicable and self-averaging is not self-evident. A practical procedure for testing self-averaging behaviour of a quantity X is to study the fluctuations - [A] v and then check if... 

A quantity is not self-averaging if the corresponding R does not decay to zero. The central limit theorem would suggest Rx,n N, while a decay of Rx slower than this would be termed as weakly self-averaging. 

An immediate consequence of this is that Rx approaches a constant as AT —> oo indicating complete absence of self-averaging at the critical point in a random system. 

These predictions have been verified for various types relevant and irrelevant disorders [51-53]. Exception to such non self-averaging behaviour with relevant disorder occurs if the Tc distribution approaches a delta function for large lattices. In such a situation, one gets back strong self-averaging behaviour [54]. 

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