In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such as the magnetic susceptibility in the ferromagnetic phase transition) described by critical exponents, universality, fractal behaviour, ergodicity breaking. Critical phenomena take place in second order phase transition, although not exclusively.
Contents
- The critical point of the 2D Ising model
- Divergences at the critical point
- Critical exponents and universality
- Critical dynamics
- Ergodicity breaking
- Mathematical tools
- Applications
- References
The critical behavior is usually different from the mean-field approximation which is valid away from the phase transition, since the latter neglects correlations, which become increasingly important as the system approaches the critical point where the correlation length diverges. Many properties of the critical behavior of a system can be derived in the framework of the renormalization group.
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
Let us consider a
where the sum is extended over the pairs of nearest neighbours and
At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below
Divergences at the critical point
The correlation length diverges at the critical point: as
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 fractal clusters the picture changes. It affects easily the smallest size clusters, since they have a nearly paramagnetic behaviour. But this change, in its turn, affects the next-scale clusters, and the perturbation climbs the ladder until the whole system changes radically. Thus, critical systems are very sensitive to small changes in the environment.
Other observables, such as the specific heat, may also diverge at this point. All these divergences stem from that of the correlation length.
Critical exponents and universality
As we approach the critical point, these diverging observables behave as
Critical dynamics
Critical phenomena may also appear for dynamic quantities, not only for static ones. In fact, the divergence of the characteristic time
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
See also superselection sector
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.
Applications
Applications arise in physics and chemistry, but also in fields such as sociology. For example, it is natural to describe a system of two political parties by an Ising model. Thereby, at a transition between one majority to the other one the above-mentioned critical phenomena may appear.