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Critical exponents describe the behaviour of physical quantities near continuous phase transitions. Remarkable about them is that they are universal, i.e. do not depend on details of the physical system, but only on
Additional recommended knowledge
These properties of critical exponents were found in experiments. The experimental results can be theoretically achieved in Mean Field Theory for higher-dimensional systems (4 or more dimensions). The theoretical treatment of lower-dimensional systems (1 or 2 dimensions) is more difficult and requires the Renormalization group.
Phase transitions occour at a certain temperature, called the critical temperature Tc. We want to describe the behaviour of a physical quantity f in terms of a power law around the critical temperature. So we introduce the reduced temperature τ: = (T − Tc) / Tc, which is zero at the phase transition, and define the critical exponent k.
This results in the power law we were looking for.
The most important critical exponents
Above and below Tc the system has two different phases characterized by an order parameter Ψ, which vanishes at and above Tc.
Let us consider the disordered phase (τ > 0), ordered phase (τ < 0 ) and critical temperature (τ = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. We have spontaneous symmetry breaking in the ordered phase. So, we will arbitrarily take any solution in the phase.
The following entries are evaluated at J = 0 (except for the δ entry)
These relations are accurate close to the critical point in two- and three-dimensional systems. In four dimensions, however, the power laws are modified by logarithmic factors. This problem does not appear in 3.99 dimensions, though.
The classical (Landau theory aka mean field theory) values are
If we add derivative terms turning it into a mean field Landau-Ginzburg theory, we get
Critical exponents are denoted by Greek letters. They fall into universality classes and obey scaling relations such as
The critical exponents can be computed from conformal field theory.
See also anomalous scaling dimension.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Critical_exponent". A list of authors is available in Wikipedia.|