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.

Definition

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''sup>2 → 0 as ''N''→∞, where ''V''_''X'' =  'X''²nbsp;−  'X''sup>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.

Non self-averaging systems

At the pure critical point randomness is classified as relevant 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. It has been shown by recent renormalization group and numerical studies that self-averaging property is lost if randomness or disorder is relevant. Most importantly as N → ∞, R_X at the critical point approaches a constant. Such systems are called non self-averaging. Thus unlike the self-averaging scenario, numerical simulations cannot lead to an improved picture in larger lattices (large N), even if the critical point is exactly known. In summary, various types of self-averaging can be indexed with the help of the asymptotic size dependence of a quantity like R_X. If R_X falls off to zero with size, it is self-averaging whereas if R_X approaches a constant as N → ∞, the system is non-self-averaging.

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 < ''z'' < 1. Such systems are classified weakly self-averaging. The known critical exponents of the system determine the exponent ''z''.

It must also be added that relevant randomness does not necessarily imply non self-averaging, especially in a mean-field scenario.
{{cite journal
, author = - S Roy and SM Bhattacharjee
, year = 2006
, title = Is small-world network disordered?
, journal = Physics Letters A
, volume = 352
, issue = 1–2
, pages = 13–16
, doi = 10.1016/j.physleta.2005.10.105
, id =
, url =
, format =
, accessdate =
, bibcode = 2006PhLA..352...13R , arxiv = cond-mat/0409012 , s2cid = 119529257

The RG arguments mentioned above need to be extended to situations with sharp limit of ''T''_''c'' distribution and long range interactions.

References

Statistical mechanics