Critical phenomena
In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the
The critical behavior is usually different from the
In order to explain the physical origin of these phenomena, we shall use the Ising model as a pedagogical example.
The critical point of the 2D Ising model
Consider a square array of classical spins which may only take two positions: +1 and −1, at a certain temperature , interacting through the Ising classical Hamiltonian:
where the sum is extended over the pairs of nearest neighbours and is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the
At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below , the state is still globally magnetized, but clusters of the opposite sign appear. As the temperature increases, these clusters start to contain smaller clusters themselves, in a typical Russian dolls picture. Their typical size, called the
Divergences at the critical point
The correlation length diverges at the critical point: as , . This divergence poses no physical problem. Other physical observables diverge at this point, leading to some confusion at the beginning.
The most important is susceptibility. Let us apply a very small magnetic field to the system in the critical point. A very small magnetic field is not able to magnetize a large coherent cluster, but with these
Other observables, such as the
Critical exponents and universality
As we approach the critical point, these diverging observables behave as for some exponent where, typically, the value of the exponent α is the same above and below Tc. These exponents are called
Critical dynamics
Critical phenomena may also appear for dynamic quantities, not only for static ones. In fact, the divergence of the characteristic time of a system is directly related to the divergence of the thermal correlation length by the introduction of a dynamical exponent z and the relation .[2] The voluminous static universality class of a system splits into different, less voluminous dynamic universality classes with different values of z but a common static critical behaviour, and by approaching the critical point one may observe all kinds of slowing-down phenomena. The divergence of relaxation time at criticality leads to singularities in various collective transport quantities, e.g., the interdiffusivity, shear viscosity ,[3] and bulk viscosity . The dynamic critical exponents follow certain scaling relations, viz., , where d is the space dimension. There is only one independent dynamic critical exponent. Values of these exponents are dictated by several universality classes. According to the Hohenberg−Halperin nomenclature,[4] for the model H[5] universality class (fluids) .
Ergodicity breaking
Ergodicity is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities. In an Ising ferromagnet below this does not happen. If , never mind how close they are, the system has chosen a global magnetization, and the phase space is divided into two regions. From one of them it is impossible to reach the other, unless a magnetic field is applied, or temperature is raised above .
See also
Mathematical tools
The main mathematical tools to study critical points are renormalization group, which takes advantage of the Russian dolls picture or the self-similarity to explain universality and predict numerically the critical exponents, and variational perturbation theory, which converts divergent perturbation expansions into convergent strong-coupling expansions relevant to critical phenomena. In two-dimensional systems, conformal field theory is a powerful tool which has discovered many new properties of 2D critical systems, employing the fact that scale invariance, along with a few other requisites, leads to an infinite symmetry group.
Critical point in renormalization group theory
The critical point is described by a
In systems in equilibrium, the critical point is reached only by precisely tuning a control parameter. However, in some non-equilibrium systems, the critical point is an attractor of the dynamics in a manner that is robust with respect to system parameters, a phenomenon referred to as self-organized criticality.[6]
Applications
Applications arise in
See also
- Ising model
- Catastrophe theory
- Critical point
- Critical exponent
- Critical opalescence
- Variational perturbation theory
- Conformal field theory
- Ergodicity
- Self-organized criticality
- Rushbrooke inequality
- Widom scaling
- Critical brain hypothesis
Bibliography
- Phase Transitions and Critical Phenomena, vol. 1-20 (1972–2001), Academic Press, Ed.: C. Domb, M.S. Green, J.L. Lebowitz
- J.J. Binney et al. (1993): The theory of critical phenomena, Clarendon press.
- N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group, Addison-Wesley.
- ISBN 981-02-4659-5 (Read online at [1])
- ISBN 0-19-851730-0
- M.E. Fisher, Renormalization Group in Theory of Critical Behavior, Reviews of Modern Physics, vol. 46, p. 597-616 (1974)
- H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
References
- .
- ^ P. C. Hohenberg und B. I. Halperin, Theory of dynamic critical phenomena , Rev. Mod. Phys. 49 (1977) 435.
- S2CID 37016085.
- S2CID 122636335.
- ISSN 0305-4470.
- ISBN 1-86094-504-X.
- ISBN 0-486-45027-9
External links
- Media related to Critical phenomena at Wikimedia Commons