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. 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. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... In physics, a critical point is the point of termination of a phase equilibrium curve, which separates two distinct phases. ... The correlation function in statistical mechanics is measure of the order in a system. ... The term scaling can have several manings: Scaling can be defined as the determination of the interdependency of variables in a physical system. ... A power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ... In electrical engineering, the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. ... Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. ... Critical Exponents are observed in second-order phase transitions. ... In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... In physics, a phase transition is the transformation of a thermodynamic system from one phase to another. ...

The critical behavior is often 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. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (Gaussian field theory, 1D Ising model. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

In order to explain the physical origin of these phenomena, we shall use the Ising model as a pedagogical example. The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ...

The critical point of the 2D Ising model

Let us consider a 2D square array of classical spins which may only take two positions: +1 and -1, at a certain temperature T, interacting through the Ising classical hamiltonian: Ernst Ising (born May 10, 1900, Cologne, Germany – May 11, 1998, Peoria, Illinois, USA) was a German physicist, who is best remembered for the development of the Ising model of ferromagnetism. ... In physics, Hamiltonian has distinct but closely related meanings. ...

where the sum is extended over the pairs of nearest neighbours and J is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the Curie temperature or critical temperature, T_c below which the system presents ferromagnetic long range order. Above it, it is paramagnetic and is apparently disordered. In physics, the Curie point, or Curie temperature, is the temperature above which a ferromagnet loses its ferromagnetic ability to possess a net (spontaneous) magnetization in the absence of an external magnetic field. ... The critical temperature, Tc, of a material is the temperature above which distinct liquid and gas phases do not exist. ... Ferromagnetism is a phenomenon by which a material can exhibit a spontaneous magnetization, and is one of the strongest forms of magnetism. ... Paramagnetism is the tendency of the atomic magnetic dipoles, due to quantum-mechanical spin, in a material that is otherwise non-magnetic to align with an external magnetic field. ...

At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below T_c, the state is still globally magnetized, but clusters of the opposite sign appears. As the temperature increases, these clusters start to contain smaller clusters themselves, in a typical Russian dolls picture. Their typical size, called the correlation length, ξ grows with temperature until it diverges at T_c. This means that the whole system is such a cluster, and there is no global magnetization. Above that temperature, the system is globally disordered, but with ordered clusters within it, whose size is again called correlation length, but it is now decreasing with temperature. At infinite temperature, it is again zero, with the system fully disordered. The correlation function in statistical mechanics is measure of the order in a system. ...

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 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. In physics, the susceptibility of a material or substance describes its response to an applied field. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... Paramagnetism is the tendency of the atomic magnetic dipoles, due to quantum-mechanical spin, in a material that is otherwise non-magnetic to align with an external magnetic field. ...

Other observables, such as the specific heat, may also diverge at this point. All these divergences stem from that of the correlation length. The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ...

Critical exponents and universality

As we approach the critical point, these diverging observables behave as for some exponent α. These exponents are called critical exponents and are robust observables. Even more, they take the same values for very different physical systems. This intriguing phenomenon, called universality is explained successfully by the renormalization group. Critical Exponents are observed in second-order phase transitions. ... See also: universalism; Self-organization, Complexity General study of systems Universality is a meta-theory arguing that ostensibly discrete systems are part of a larger complex system that extends across several scales (spatially and temporally), and emerges in patterns during criticality. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

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 T_c this does not happen. If T < T_c, 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 T_c. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ...

See also superselection sector A superselection sector is a concept used in quantum mechanics. ...

Mathematical tools

The main mathematical tools to study critical points are renormalization group, which takes advantage of the Russian dolls picture 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. In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... Mathematical method to convert divergent power series in a small expansion parameter, say , into convergent series in powers , where is a critical exponent. ... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... The symmetry group of an object (e. ...

See also

• Ising model
• Critical point
• Critical exponent
• Renormalization group
• Variational perturbation theory
• Conformal field theory
• Ergodicity
• Self-organized criticality

The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... The term critical point can mean any of: critical point (thermodynamics) critical point (mathematics) critical loops (topology) critical point (set theory) This is a disambiguation page: a list of articles associated with the same title. ... Critical Exponents are observed in second-order phase transitions. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... Mathematical method to convert divergent power series in a small expansion parameter, say , into convergent series in powers , where is a critical exponent. ... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... The theory of self-organized criticality (SOC) claims that whenever a self-organizing dynamical system is open or dissipative, it exhibits critical (scale-invariant) behaviour similar to that displayed by static systems undergoing a second-order phase transition. ...

Bibliography

• 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.
• H. Kleinert and V. Schulte Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-76 (Read online at [1])