Self-averaging

A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz.

A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz. Frequently in physics one comes across situations where quenched randomness plays an important role. Any physical property X of such a system, would require an averaging over all disorder realisations. The system can be completely described by the average [X] where [...] denotes averaging over realisations (“averaging over samples”) provided the relative variance R X  = V X  / [X] 2  → 0 as N→∞, where V X  = [X 2 ] − [X] 2 and N denotes the size of the realisation. In such a scenario a single large system is sufficient to represent the whole ensemble. Such quantities are called self-averaging. Away from criticality, when the larger lattice is built from smaller blocks, then due to the additivity property of an extensive quantity, the central limit theorem guarantees that R X  ~ N −1 thereby ensuring self-averaging. On the other hand, at the critical point, the question whether X is self-averaging or not becomes nontrivial, due to long range correlations.

Strong and weak self-averaging

There is a further classification of self-averaging systems as strong and weak. If the exhibited behavior is R X  ~ N −1 as suggested by the central limit theorem, mentioned earlier, the system is said to be strongly self-averaging. Some systems shows a slower power law decay R X  ~ N −z with 0  c distribution and long range interactions.

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