Editing Order and disorder (section)

===Long-range order===

'''Long-range order''' characterizes physical [[system]]s in which remote portions of the same sample exhibit [[correlation|correlated]] behavior.

This can be expressed as a [[correlation function]], namely the [[Spin (physics)|spin-spin correlation function]]:

: <math>G(x,x') = \langle s(x),s(x') \rangle. \, </math>

where ''s'' is the spin quantum number and ''x'' is the distance function within the particular system.

This function is equal to unity when <math>x=x'</math> and decreases as the distance <math>|x-x'|</math> increases.  Typically, it [[exponential decay|decays exponentially]] to zero at large distances, and the system is considered to be disordered. But if the correlation function decays to a constant value at large <math>|x-x'|</math> then the system is said to possess long-range order.  If it decays to zero as a power of the distance then it is called quasi-long-range order (for details see Chapter 11 in the textbook cited below. See also [[Kosterlitz-Thouless transition|Berezinskii–Kosterlitz–Thouless transition]]). Note that what constitutes a large value of <math>|x-x'|</math> is understood in the sense of [[asymptotic analysis|asymptotics]].