World Library  
Flag as Inappropriate
Email this Article

Critical exponent

Article Id: WHEBN0002029677
Reproduction Date:

Title: Critical exponent  
Author: World Heritage Encyclopedia
Language: English
Subject: Variational perturbation theory, Critical phenomena, Power law, Phase transition, Richard Feynman
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Critical exponent

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 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.

Definition

Phase transitions occur at a certain temperature, called the critical temperature T_c. 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 \tau := (T-T_c)/T_c, which is zero at the phase transition, and define the critical exponent k.

k \, \stackrel{\text{def}}{=} \, \lim_{\tau \to 0}{\log |f(\tau)| \over \log |\tau|} \text{.}

This results in the power law we were looking for.

f(\tau) \propto \tau^k, \quad \tau\approx 0 \text{.}

It is important to remember that this represents the asymptotic behavior of the function f(\tau) as \tau \to 0.

More generally one might expect

f(\tau)=A \tau^k (1+b\tau ^{k_1} + \cdots) \text{.}

The most important critical exponents

Below T_c the system has two different phases characterized by an order parameter \Psi, which vanishes at and above T_c.

Let us consider the disordered phase (\tau > 0), ordered phase (\tau < 0 ) and critical temperature (\tau = 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.
Keys
\Psi order parameter (e.g. \frac{\rho-\rho_c}{\rho_c} for the liquid-gas critical point, magnetization for the Curie point,etc.)
\tau \frac{T-T_c}{T_c}
f specific free energy
C specific heat; -T\frac{\partial^2 f}{\partial T^2}
J source field (e.g. \frac{P-P_c}{P_c} where P is the pressure and Pc the critical pressure for the liquid-gas critical point, reduced chemical potential, the magnetic field H for the Curie point )
\chi the susceptibility/compressibility/etc.; \frac{\partial \Psi}{\partial J}
\xi correlation length
d the number of spatial dimensions
\left\langle \psi(\vec{x}) \psi(\vec{y}) \right\rangle the correlation function
r spatial distance
The following entries are evaluated at J = 0 (except for the \delta entry)
Critical exponents for \tau > 0 (disordered phase)
Greek letter relation
\alpha C \propto \tau^{-\alpha}
\gamma \chi \propto \tau^{-\gamma}
\nu \xi \propto \tau^{-\nu}
Critical exponents for \tau < 0 (ordered phase)
Greek letter relation
\alpha^\prime C \propto (-\tau)^{-\alpha^\prime}
\beta \Psi \propto (-\tau)^{\beta}
\gamma^\prime \chi \propto (-\tau)^{-\gamma^\prime}
\nu^\prime \xi \propto (-\tau)^{-\nu^\prime}
Critical exponents for \tau = 0
Greek letter relation
\delta J \propto \Psi^\delta
\eta \left\langle \psi(0) \psi(r) \right\rangle \propto r^{-d+2-\eta}

The critical exponents can be derived from the specific free energy f(J,T) as a function of the source and temperature. The correlation length can be derived from the functional F[J;T].

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.

Mean field critical exponents of Ising-like systems

The classical Landau theory (aka mean field theory) values of the critical exponents for a scalar field (of which the Ising-model is the prototype example) are given by

\alpha = \alpha^\prime = 0\,
\beta = \frac{1}{2}\,
\gamma = \gamma^\prime = 1\,
\delta = 3\,

If we add derivative terms turning it into a mean field Ginzburg–Landau theory, we get

\eta = 0\,
\nu = \frac{1}{2}\,

One of the major discoveries in the study of critical phenomena is that mean field theory of critical points is only correct when the space dimension of the system is four or higher (which unfortunately excludes many of the experimentally relevant cases). This dimension is called the upper critical dimension. The problem with mean field theory is that the critical exponents do not depend on the space dimension. This leads to a quantitative discrepancy in space dimensions 2 and 3, where the true critical exponents differ from the mean field values. It leads to a qualitative discrepancy in space dimension 1, where a critical point in fact no longer exists, even though mean field theory still predicts there is one. The space dimension where mean field theory becomes qualitatively incorrect is called the lower critical dimension.

Experimental values

The most accurately measured value of \alpha is −0.0127(3) for the phase transition of superfluid helium (the so-called lambda transition). The value was measured on a space shuttle to minimize pressure differences in the sample (see here). Interestingly, this value is in a significant disagreement with the most precise theoretical determination by a combination of Monte Carlo and high temperature expansion techniques. Other techniques give results in agreement in the experiment but are less precise. See Table 2 in this review article.

Scaling functions

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 a^\Delta for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.

Scaling relations

\alpha \equiv \alpha'
\gamma \equiv \gamma'
\nu \equiv \nu'

Thus, the exponents above and below the critical temperature, respectively, have identical values. This is understandable, since the respective scaling functions, f_\pm(k\xi ,\dots), originally defined for k\xi \ll 1, should become identical if extrapolated to k\xi \gg 1\,.

Critical exponents are denoted by Greek letters. They fall into universality classes and obey the scaling relations

\nu d = 2 - \alpha = 2\beta + \gamma = \beta(\delta + 1) = \gamma \frac{\delta + 1}{\delta - 1}\,
2 - \eta = \frac{\gamma}{\nu} = d \frac{\delta - 1}{\delta + 1}

These equations imply that there are only two independent exponents, e.g., \,\nu and \eta\,. All this follows from the theory of the renormalization group.

Anisotropy

There are some anisotropic systems where the correlation length is direction dependent.

Multicritical points

More complex behaviour may occur at multicritical points, at the border or on intersections of critical manifolds.

Static versus dynamic properties

The above examples exclusively refer to the static properties of a critical system. However dynamic properties of the system may become critical, too. Especially, the characteristic time, \tau_{\mathrm{char}}, of a system diverges as \tau_{\mathrm{char}}\propto \xi^z, with a dynamical exponent z. Moreover, the large static universality classes of equivalent models with identical static critical exponents decompose into smaller dynamical universality classes, if one demands that also the dynamical exponents are identical.

The critical exponents can be computed from conformal field theory.

See also anomalous scaling dimension.

Transport properties

Critical exponents also exist for transport quantities like viscosity and heat conductivity.

Self-organized criticality

Critical exponents also exist for self organized criticality for dissipative systems.

Percolation Theory

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.

See also

External links and literature

  • Hagen Kleinert and Verena Schulte-Frohlinde, -Theories4Critical Properties of φ, 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. E. Stanley Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, 1971
  • A. Bunde and S. Havlin (editors), Fractals in Science, Springer, 1995
  • A. Bunde and S. Havlin (editors), Fractals and Disordered Systems, Springer, 1996
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.