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'']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 <math>X</math> 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.<ref>{{cite journal | author = -A. Aharony and A.B. Harris | year = 1996 | title = Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points | journal = Phys. Rev. Lett. | volume = 77 | issue = 18 | pages = 3700–3703 | doi = 10.1103/PhysRevLett.77.3700 | id = | url = http://repository.upenn.edu/cgi/viewcontent.cgi?article=1473&context=physics_papers| format = | accessdate = | pmid = 10062286 | bibcode=1996PhRvL..77.3700A}}</ref> Most importantly as N → ∞, RX 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 RX. If RX falls off to zero with size, it is self-averaging whereas if RX 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. <ref>{{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 }}</ref> The RG arguments mentioned above need to be extended to situations with sharp limit of ''T''''c'' distribution and long range interactions.

== References ==

Category:Statistical mechanics