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{{About|physical systems|the property of an infinite word|Critical exponent of a word}}
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'''Critical exponents''' describe the behaviour of physical quantities near continuous [[phase transitions]]. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on
* the dimension of the system,
* the range of the interaction,
* the [[Spin (physics)|spin]] dimension.
These properties of critical exponents are supported by experimental data. 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 and critical exponents appear also in percolation systems.


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[[Phase transitions]] occur at a certain temperature, called the [[critical temperature]] <math>T_c</math>. We want to describe the behaviour of a physical quantity <math>f</math> in terms of a [[power law]] around the critical temperature. So we introduce the [[reduced temperature]] <math>\tau := (T-T_c)/T_c</math>, which is zero at the [[phase transition]], and define the critical exponent <math>k</math>.
 
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:<math>k \, \stackrel{\text{def}}{=} \, \lim_{\tau \to 0}{\log |f(\tau)| \over \log |\tau|} \text{.}</math>
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:<math> f(\tau) \propto \tau^k, \quad \tau\approx 0 \text{.}</math>
 
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It is important to remember that this  represents the asymptotic behavior of the function <math>f(\tau)</math> as <math>\tau \to 0</math>.
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:<math>f(\tau)=A \tau^k (1+b\tau ^{k_1} + \cdots) \text{.}</math>
 
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==The most important critical exponents==
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Below <math>T_c</math> the system has two different phases characterized by an [[order parameter]] <math>\Psi</math>, which vanishes at and above <math>T_c</math>.


Let us consider the [[disordered phase]] (<math>\tau</math> > 0), [[ordered phase]] (<math>\tau</math> < 0 ) and [[critical temperature]] (<math>\tau</math> = 0) phases separately. Following the standard convention, the critical exponents related to the ordered phase are primed. It's also another standard convention to use the super/subscript +(-) for the disordered(ordered) state. We have [[spontaneous symmetry breaking]] in the ordered phase. So, we will arbitrarily take any solution in the phase.
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{| style="border:#ddd 1px solid"
 
! colspan=2 | Keys
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|-
 
| <math>\Psi</math>
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| order parameter  (e.g. <math> \frac{\rho-\rho_c}{\rho_c} </math> for the liquid-gas critical point, [[magnetization]] for the [[Curie point]],etc.)
 
|-
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| <math>\tau</math>
 
| <math>\frac{T-T_c}{T_c}</math>
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|-
 
| <math>f</math>
</ul>
| specific [[Thermodynamic free energy|free energy]]
|-
| <math>C</math>
| [[specific heat]]; <math>-T\frac{\partial^2 f}{\partial T^2}</math>
|-
| <math>J</math>
| source field (e.g. <math>\frac{P-P_c}{P_c}</math> where ''P'' is the [[pressure]] and ''P''<sub>c</sub> the [[critical pressure]] for the liquid-gas critical point,  reduced [[chemical potential]], the [[magnetic field]] ''H'' for the [[Curie point]] )
|-
| <math>\chi</math>
| the [[susceptibility]]/[[compressibility]]/etc.; <math>\frac{\partial \Psi}{\partial J}</math>
|-
| <math>\xi</math>
| [[correlation length]]
|-
| <math>d</math>
| the number of spatial [[dimension]]s
|-
| <math>\left\langle \psi(\vec{x}) \psi(\vec{y}) \right\rangle</math>
| the [[correlation function (statistical mechanics)|correlation function]]
|-
| <math>r</math>
| spatial distance
|}


The following entries are evaluated at <math>J = 0</math> (except for the <math>\delta</math> entry)
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{| style="border:#ddd 1px solid"
|-
! colspan=2 | Critical exponents for <math>\tau</math> > 0 (disordered phase)
|-
| Greek letter
| relation
|-
| <math>\alpha</math>
| <math>C \propto \tau^{-\alpha}</math>
|-
| <math>\gamma</math>
| <math>\chi \propto \tau^{-\gamma}</math>
|-
| <math>\nu</math>
| <math>\xi \propto \tau^{-\nu}</math></sup>
|-
|-
! colspan=2 | Critical exponents for <math>\tau</math> < 0 (ordered phase)
|-
| Greek letter
| relation
|-
| <math>\alpha^\prime</math>
| <math>C \propto (-\tau)^{-\alpha^\prime}</math>
|-
| <math>\beta</math>
| <math>\Psi \propto (-\tau)^{\beta}</math>
|-
| <math>\gamma^\prime</math>
| <math>\chi \propto (-\tau)^{-\gamma^\prime}</math>
|-
| <math>\nu^\prime</math>
| <math>\xi \propto (-\tau)^{-\nu^\prime}</math>
|-
|-
! Critical exponents for <math>\tau</math> = 0
|-
| Greek letter
| relation
|-
| <math>\delta</math>
| <math>J \propto \Psi^\delta</math>
|-
| <math>\eta</math>
| <math>\left\langle \psi(0) \psi(r) \right\rangle \propto r^{-d+2-\eta}</math>
|}


The critical exponents can be derived from the specific free energy <math>f(J,T)</math> as a function of the source and temperature. The correlation length can be derived from the [[functional (mathematics)|functional]] <math>F[J;T]</math>.
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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.
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This problem does not appear in [[dimensional regularization|3.99 dimensions]], though. <!-- :) -->


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If we add derivative terms turning it into a mean field [[Ginzburg–Landau theory]], we get
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:<math>\eta = 0\,</math>
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In light of the critical scalings, we can reexpress all thermodynamic quantities in terms of dimensionless quantities. Close enough to the critical point, everything can be reexpressed in terms of certain ratios of the powers of the reduced quantities. These are the scaling functions.


The origin of scaling functions can be seen from the renormalization group. The critical point is an [[infrared fixed point]]. In a sufficiently small neighborhood of the critical point, we may linearize the action of the renormalization group. This basically means that rescaling the system by a factor of ''a'' will be equivalent to rescaling operators and source fields by a factor of <math>a^\Delta</math> for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.
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:<math>\alpha \equiv \alpha'</math>
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Thus, the exponents above and below the critical temperature, respectively,  have identical values. This is understandable, since the respective scaling functions, <math> f_\pm(k\xi ,\dots)</math>, originally defined for <math>k\xi \ll 1</math>, should become identical if extrapolated to <math> k\xi \gg 1\,.</math>
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Critical exponents are denoted by Greek letters. They fall into [[universality classes]] and obey the [[scaling relation]]s
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:<math> \nu d = 2 - \alpha = 2\beta + \gamma = \beta(\delta + 1) = \gamma \frac{\delta + 1}{\delta - 1}\,</math>
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:<math> 2 - \eta = \frac{\gamma}{\nu} = d \frac{\delta - 1}{\delta + 1}</math>
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These equations imply that there are only two independent exponents, e.g., <math>\,\nu</math> and {{nowrap|<math>\eta\,</math>.}} All this follows from the theory of the [[renormalization group]].
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More complex behaviour may occur at [[multicritical point]]s, at the border or on intersections of critical manifolds.
 
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The critical exponents can be computed from [[conformal field theory]].
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See also [[anomalous scaling dimension]].
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Critical exponents also exist for transport quantities like [[viscosity]] and [[heat conductivity]].
 
 
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== Self-organized criticality ==
 
Critical exponents also exist for self organized criticality for [[dissipative system]]s.
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Phase transitions and critical exponents appear also in percolation processes where the concentration of occupied sites or links play the role of temperature. See [[percolation critical exponents]].
 
 
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== See also ==
 
* [[Rushbrooke inequality]]
</ul>
* [[Widom scaling]]
* [[Ising critical exponents]]
 
==External links and literature==
* [[Hagen Kleinert]] and Verena Schulte-Frohlinde, ''[http://www.physik.fu-berlin.de/~kleinert/b8 Critical Properties of φ<sup>4</sup>-Theories]'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapore, 2001)];  Paperback ISBN 981-02-4658-7
* Toda, M., Kubo, R., N. Saito, ''Statistical Physics I'', Springer-Verlag (Berlin, 1983); Hardcover ISBN 3-540-11460-2
* J.M.Yeomans, ''Statistical Mechanics of Phase Transitions'', Oxford Clarendon Press
* [[H. Eugene Stanley|H. E. Stanley]] ''Introduction to Phase Transitions and Critical Phenomena'', Oxford University Press, 1971
* A. Bunde and [[Shlomo Havlin|S. Havlin]] (editors), ''[http://havlin.biu.ac.il/Shlomo%20Havlin%20books_f_in_s.php Fractals in Science]'', Springer, 1995
* A. Bunde and [[Shlomo Havlin|S. Havlin]] (editors), ''[http://havlin.biu.ac.il/Shlomo%20Havlin%20books_fds.php Fractals and Disordered Systems]'', Springer, 1996
 
[[Category:Phase transitions]]
[[Category:Critical phenomena]]
[[Category:Renormalization group]]

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