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 transitions
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 (crystalline ordering), rotational
(ferroelectric ordering), or a spin state
(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.
== Characterizing 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 lattices
and 230 space groups.
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 quasicrystals
in 1982 showed that there are perfectly deterministic
tilings that do not possess lattice periodicity.
Besides structural order, one may consider
charge ordering, spin ordering, magnetic ordering,
and compositional ordering. Magnetic ordering
is observable in neutron diffraction.
It is a thermodynamic 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 energy.
=== Long-range order ===
Long-range order characterizes physical systems
in which remote portions of the same sample
exhibit correlated behavior.
This can be expressed as a correlation function,
namely the spin-spin correlation function:
G
(
x
,
x
′
)
=
⟨
s
(
x
)
,
s
(
x
′
)
⟩
.
{\displaystyle G(x,x')=\langle s(x),s(x')\rangle
.\,}
where s is the spin quantum number and x is
the distance function within the particular
system.
This function is equal to unity when
x
=
x
′
{\displaystyle x=x'}
and decreases as the distance
|
x
−
x
′
|
{\displaystyle |x-x'|}
increases. Typically, it 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
|
x
−
x
′
|
{\displaystyle |x-x'|}
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 Berezinskii–Kosterlitz–Thouless
transition). Note that what constitutes a
large value of
|
x
−
x
′
|
{\displaystyle |x-x'|}
is understood in the sense of asymptotics.
== 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 quenched or frozen. Spin glasses 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 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.
== Annealed disorder ==
A system is said to present annealed disorder
when some parameters entering its definition
are random variables, but whose evolution
is related to that of the degrees of freedom
defining the system. It is defined in opposition
to quenched disorder, where the random variables
may not change their values.
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: Gauge Fields in Condensed Matter
(ISBN 9971-5-0210-0, 2 volumes) Singapore:
World Scientific (1989).
