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 Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

one comes across situations where quenched randomness plays an important role. Any

physical property A physical property is any property of a physical system that is measurable. The changes in the physical properties of a system can be used to describe its changes between momentary states. A quantifiable physical property is called ''physical ...

''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 Physical or chemical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. The terms "intensive and extensive ...

, the

central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...

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

correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...

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 In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...

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