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SelfaveragingSelfaveraging is a property of a system where a physical property of a complete disordered system can be described by averaging over a sufficiently large sample; it was introduced by Ilya Mikhailovich Lifshitz. Additional recommended knowledge
DefinitionVery often in physics one comes across situations where quenched randomness plays an important role. Therefore, any physical property X of such a disordered system, would require an averaging over all realisations. It would suffice to have a description in terms of the average [X] where [...] denotes an averaging over realisations (“averaging over samples”) provided the relative variance R_{X} = V_{X}/[X]^{2} → 0 for large X, where V_{X} = [X^{2}] − [X]^{2}. In such a case a single large system is enough to represent the whole ensemble. Such quantities are called selfaveraging. Off criticality, when one builds up a large lattice from smaller blocks, then due to the additivity property of an extensive quantity, central limit theorem guarantees that R_{X} → N^{−1} ensuring selfaveraging. In contrast, at a critical point, due to long range correlations the answer whether X is selfaveraging or not becomes nontrivial. Non selfaveraging systemsRandomness at a pure critical point is classified as relevant or irrelevant if, by the standard definition of relevance, it leads to a change in the critical behaviour (i.e., the critical exponents) of the pure system. Recent renormalization group and numerical studies have shown that if randomness or disorder is relevant, then selfaveraging property is lost ^{[1]}. In particular, R_{X} at the critical point approaches a constant as N → ∞. Such systems are called non selfaveraging. A serious consequence of this is that unlike the selfaveraging case, even if the critical point is known exactly, statistics in numerical simulations cannot be improved by going over to larger lattices (large N). Let us recollect the definitions of various types of selfaveraging with the help of the asymptotic size dependence of a quantity like R_{X}. If R_{X} approaches a constant as N → ∞, the system is nonselfaveraging while if R_{X} decays to zero with size, it is selfaveraging. Strong and weak selfaveragingSelfaveraging systems are further classified as strong and weak. If the decay is R_{X} ~ N^{−1} as suggested by the central limit theorem, mentioned earlier, the system is said to be strongly selfaveraging. There is yet another class of systems which shows a slower power law decay R_{X} ~ N^{−z} with 0 < z < 1. Such cases are known as weakly selfaveraging. The exponent z is determined by the known critical exponents of the system. It must also be added that relevant randomness does not necessarily imply non selfaveraging, especially in a meanfield scenario ^{[2]} . An extension of the RG arguments mentioned above to encompass situations with sharp limit of T_{c} distribution and long range interactions, may shed light on this. References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Selfaveraging". A list of authors is available in Wikipedia. 