Self-averaging

Self-averaging 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.

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Definition

Very 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 = [X²] − [X]². In such a case a single large system is enough to represent the whole ensemble. Such quantities are called self-averaging. 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 self-averaging. In contrast, at a critical point, due to long range correlations the answer whether X is self-averaging or not becomes nontrivial.

Non self-averaging systems

Randomness 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 self-averaging property is lost ^[1]. In particular, R_X at the critical point approaches a constant as N → ∞. Such systems are called non self-averaging. A serious consequence of this is that unlike the self-averaging 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 self-averaging 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 non-self-averaging while if R_X decays to zero with size, it is self-averaging.

Strong and weak self-averaging

Self-averaging 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 self-averaging. 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 self-averaging. 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 self-averaging, especially in a mean-field 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

1. ^ -A. Aharony and A.B. Harris (1996). "Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points". Phys. Rev. Lett. 77. doi:10.1103/PhysRevLett.77.3700.
2. ^ - Soumen Roy and Somendra M. Bhattacharjee (2006). "Is small-world network disordered?". Physics Letters A 352. doi:10.1016/j.physleta.2005.10.105.