Critical exponents describe the behavior 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 some of its general features. For instance, for ferromagnetic systems, the critical exponents depend 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. Analytical results can be theoretically achieved in mean field theory in high dimensions or when exact solutions are known such as the two-dimensional Ising model. The theoretical treatment in generic dimensions requires the renormalization group approach. Phase transitions and critical exponents appear in many physical systems such as water at the liquid-vapor transition, in ferro or antiferromagnetic systems, in superconductivity, in percolation, in systems of particles that diffuse and undergo chemical reactions, in turbulent fluids,.... The critical dimension above which mean field exponents are valid varies with the systems and can even be infinite. It is 4 for the liquid-vapor transition, 6 for percolation and probably infinite for turbulence. Mean field critical exponents are also valid for random graphs, such as Erdős–Rényi graphs, which can be regarded as infinite dimensional systems.
- 1 Definition
- 2 The most important critical exponents
- 3 Mean field critical exponents of Ising-like systems
- 4 Experimental values
- 5 Scaling functions
- 6 Scaling relations
- 7 Anisotropy
- 8 Multicritical points
- 9 Static versus dynamic properties
- 10 Transport properties
- 11 Self-organized criticality
- 12 Percolation Theory
- 13 See also
- 14 External links and literature
- 15 References
The control parameter that drives phase transitions is often temperature but can also be other macroscopic variables like pressure or an external magnetic field. For simplicity, the following discussion works in terms of temperature; the translation to another control prameter is straightforward. The temperature at which the transition occurs is called the critical temperature Tc. We want to describe the behavior of a physical quantity f in terms of a power law around the critical temperature; we introduce the reduced temperature
which is zero at the phase transition, and define the critical exponent :
This results in the power law we were looking for:
It is important to remember that this represents the asymptotic behavior of the function f(τ) as τ → 0.
More generally one might expect
The most important critical exponents
Let us assume that the system has two different phases characterized by an order parameter Ψ, which vanishes at and above Tc.
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. It is also another standard convention to use superscript/subscript + (−) for the disordered (ordered) state. In general spontaneous symmetry breaking occurs in the ordered phase.
|Ψ||order parameter (e.g. ρ − ρc/ for the liquid–gas critical point, magnetization for the Curie point, etc.)|
|τ||T − Tc/|
|f||specific free energy|
|C||specific heat; −T∂2f/|
|J||source field (e.g. P − Pc/ 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)|
|χ||the susceptibility, compressibility, etc.; ∂ψ/|
|d||the number of spatial dimensions|
|⟨ψ(x→) ψ(y→)⟩||the correlation function|
The following entries are evaluated at J = 0 (except for the δ entry)
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. These do not appear in dimensions arbitrarily close to but not exactly four, which can be used as a way around this problem.
Mean field critical exponents of Ising-like systems
If we add derivative terms turning it into a mean field Ginzburg–Landau theory, we get
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 higher than a certain dimension called the upper critical dimension which excludes the physical dimensions 1, 2 or 3 in most cases. The problem with mean field theory is that the critical exponents do not depend on the space dimension. This leads to a quantitative discrepancy below the critical dimensions, where the true critical exponents differ from the mean field values. It can even lead to a qualitative discrepancy at low space dimension, where a critical point in fact can no longer exist, even though mean field theory still predicts there is one. This is the case for the Ising model in dimension 1 where there is no phase transition. The space dimension where mean field theory becomes qualitatively incorrect is called the lower critical dimension.
The most accurately measured value of α 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. 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.
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 aΔ for some Δ. So, we may reparameterize all quantities in terms of rescaled scale independent quantities.
It was believed for a long time that the critical exponents were the same above and below the critical temperature, e.g. α ≡ α′ or γ ≡ γ′. It has now been shown that this is not necessarily true: When a continuous symmetry is explicitly broken down to a discrete symmetry by irrelevant (in the renormalization group sense) anisotropies, then the exponents γ and γ′ are not identical.
These equations imply that there are only two independent exponents, e.g., ν and η. All this follows from the theory of the renormalization group.
Directed percolation can be also regarded as anisotropic percolation. In this case the critical exponents are different and the upper critical dimension is 5.
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, τchar, of a system diverges as τchar ∝ ξz, 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. For critical exponents for dynamics in percolation systems see reference.
The critical exponents can be computed from conformal field theory.
See also anomalous scaling dimension.
Critical exponents also exist for self organized criticality for dissipative systems.
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. For percolation the critical exponents are different from Ising. For example, in the mean field for percolation compared to for Ising. In network theory, the strength of interactions between communities has been found to behave analogous to an external field in magnets near the pase transition or as ghost field in percolation.
- Complex networks
- Random graphs
- Rushbrooke inequality
- Widom scaling
- Ising critical exponents
- Percolation critical exponents
- Network science
- Percolation theory
- Graph theory
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