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, and ergodicity breaking.
Critical phenomena take place in second order phase transitions, although not exclusively.
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.
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
below which the system presents ferromagnetic long range order.
Other physical observables diverge at this point, leading to some confusion at the beginning.
Let us apply a very small magnetic field to the system in the critical point.
It affects easily the smallest size clusters, since they have a nearly paramagnetic behaviour.
As we approach the critical point, these diverging observables behave as
This intriguing phenomenon, called universality, is explained, qualitatively and also quantitatively, by the renormalization group.
of a system is directly related to the divergence of the thermal correlation length
[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.
at criticality leads to singularities in various collective transport quantities, e.g., the interdiffusivity, shear viscosity
The dynamic critical exponents follow certain scaling relations, viz.,
Ergodicity is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities.
, 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 superselection sector 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.
The critical point is described by a conformal field theory.
According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite.
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.
For example, it is natural to describe a system of two political parties by an Ising model.