en>LokiClock: /* Intro */ Zassenhaus lemma

2014-01-14T13:23:37Z

Intro: Zassenhaus lemma

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{{Unreferenced|date=February 2011}}
In [[physics]], the terms '''order''' and '''disorder''' designate the presence or absence of some [[symmetry]] or [[correlation]] in a many-particle system.

In [[condensed matter physics]], systems typically are ordered at low temperatures; upon heating, they undergo one or several [[phase transition]]s into less ordered states.
Examples for such an '''order-disorder transition''' are:
* the melting of ice: solid-liquid transition, loss of crystalline order;
* the demagnetization of iron by heating above the [[Curie temperature]]: ferromagnetic-paramagnetic transition, loss of magnetic order.

The degree of freedom that is ordered or disordered can be translational ([[crystal]]line ordering), rotational ([[ferroelectricity|ferroelectric]] ordering), or a spin state ([[magnetism|magnetic]] ordering).

The order can consist either in a full crystalline [[space group]] symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of '''long-range order''' or '''short-range order'''.

If a disordered state is not in [[thermodynamic equilibrium]], one speaks of '''quenched disorder'''. For instance, a [[glass]] is obtained by quenching (supercooling) a liquid. By extension, other quenched states are called [[spin glass]], [[orientational glass]]. In some contexts, the opposite of quenched disorder is '''annealed disorder'''.

==Characterising order==

===Lattice periodicity and X-ray crystallinity===

The strictest form of order in a solid is '''lattice periodicity''': a certain pattern (the arrangement of atoms in a [[unit cell]]) is repeated again and again to form a translationally invariant [[tiling]] of space. This is the defining property of a [[crystal]]. Possible symmetries have been classified in 14 [[Bravais lattice]]s and 230 [[space group]]s.

Lattice periodicity implies '''long-range order''': if only one unit cell is known, then by virtue of the translational symmetry it is possible to accurately predict all atomic positions at arbitrary distances. During much of the 20th century, the converse was also taken for granted - until the discovery of [[quasicrystal]]s in 1982 showed that there are perfectly deterministic tilings that do not possess lattice periodicity.

Besides structural order, one may consider [[charge ordering]], [[Spin (physics)|spin]] ordering, [[magnetic ordering]], and compositional ordering. Magnetic ordering is observable in [[neutron diffraction]].

It is a [[thermodynamic]] [[Entropy (order and disorder)|entropy]] concept often displayed by a second-order [[phase transition]]. Generally speaking, high thermal energy is associated with disorder and low thermal energy with ordering, although there have been violations of this. Ordering peaks become apparent in diffraction experiments at low angles.

===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. If, however, 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 relative.

==Quenched disorder==

In [[statistical physics]], a system is said to present '''quenched disorder''' when some parameters defining its behaviour are random variables which do not evolve with time, i.e.: they are [[quenching|quenched]] or ''frozen''. [[Spin glass]]es are a typical example. It is opposite to [[annealed disorder]], where the random variables are allowed to evolve themselves.

In mathematical terms, quenched disorder is harder to analyze than its annealed counterpart, since the thermal and the noise averaging play very different roles. In fact, the problem is so hard that few techniques to approach each are known, most of them relying on approximations. The most used are 1) a technique based on a mathematical analytical continuation known as the [[replica trick]] and 2) the [[Cavity method]]; although these give results in accord with experiments in a large range of problems, they are not generally proven to be a rigorous mathematical procedure. More recently it has been shown by rigorous methods, however, that at least in the archetypal spin-glass model (the so-called [[Spin_glass#The_model_of_Sherrington_and_Kirkpatrick|Sherrington-Kirkpatrick model]]) the replica based solution is indeed exact. The second most used technique in this field is [[generating functional analysis]]. This method is based on path integrals, and is in principle fully exact, although generally more difficult to apply than the replica procedure.

[[Image:Ordering.png|thumbnail|600px|center|Transition from disordered (left) to ordered (right) states]]

==Annealed disorder==

A system is said to present '''annealed disorder''' when some parameters entering its definition are [[random variable]]s, but whose evolution is related to that of the [[Degrees of freedom (physics and chemistry)|degrees of freedom]] defining the system. It is defined in opposition to quenched disorder, where the random variables may not change its value.

Systems with annealed disorder are usually considered to be easier to deal with mathematically, since the average on the disorder and the [[thermal average]] may be treated on the same footing.

==See also==

* In high energy physics, the formation of the [[chiral condensate]] in [[quantum chromodynamics]] is an ordering transition; it is discussed in terms of [[superselection]].
* [[Entropy]]
* [[Topological order]]
* [[Impurity]]
* [[superstructure (physics)]]

== Further reading ==

* H Kleinert: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1 ''Gauge Fields in Condensed Matter''] (ISBN 9971-5-0210-0, 2 volumes) Singapore: World Scientific (1989).

[[Category:Statistical mechanics]]
[[Category:Crystallography]]

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Schreier refinement theorem - Revision history

en>LokiClock: /* Intro */ Zassenhaus lemma