Critical phenomena
Critical phenomena are a generic term for the characteristic behaviors in the physics of critical points, but, for example, in sociology ( " physical sociodynamics "). The phenomena can be mostly attributed to the divergence of a correlation length. Characteristic is also the related phenomenon of "critical slowing down ".
Quantitatively, the critical phenomena, especially by critical exponents, algebraic divergences of order parameters and scaling relationships between different variables, universality, fractal behavior and the violation of ergodicity are marked.
Critical phenomena are partly - but not exclusively - during phase transitions of the second order on. Is particularly characteristic for almost all models the divergence of the correlation length when approaching the critical temperature, ie, with a model-dependent, but consistent within a very large universality class of the critical exponents.
- 2.1 Critical dynamics
- 2.2 Critical opalescence
2D Ising model
To illustrate the behavior of critical phenomena the two-dimensional Ising model can be used. The model describes a field of classical spins, which can take only two discrete states 1 and -1. The interaction is described by the classical Hamiltonian:
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Here, the sum extends over adjacent pairs and is assumed to be constant coupling constant. If this is positive, the system below a critical temperature, the Curie temperature, ferromagnetic long-range order. Above this temperature, it is paramagnetic and temporal averaging without order.
At absolute zero can only assume one of the values 1 or -1, the thermal expectation value. At higher temperatures, the state is still magnetized below overall, with, that is, there are now entering regions ( clusters ) of different sign. The typical diameter of these clusters is called the correlation length. The clusters themselves are made smaller and smaller clusters with increasing the temperature. The correlation length increases with temperature until it diverges at. This means that the entire system now is a single cluster, and there are no more global magnetization.
Above the critical temperature, the system is globally disordered, but it consists of higher-level cluster, the size of which decreases with increasing temperature. The size of the clusters in turn defines the correlation length. In the limit of very large temperatures, these zero and the system is again completely disordered.
Critical Point
The correlation length diverges at the critical point: . This divergence is the reason that other physical quantities diverge at this point or go with special power laws to zero. In this case, the correlation length is the length scale back on there is a correlation between events, or to extend the fluctuations.
In addition to the correlation length, the susceptibility is a critical point diverging size. When the system is exposed to a small magnetic field, realized via an additional term in the Hamiltonian, so this will not be able to magnetize a large coherent cluster. If small fractal clusters exist, however, the picture changes. The smallest of these clusters are easily influenced, as they show a nearly paramagnetic behavior. However, this change also affects the next larger clusters and the disturbance spreads rapidly and radically changed the entire system. Critical systems are therefore extremely sensitive to small changes in the environment. Other sizes, such as the specific heat, may also diverge at this point.
Violation of ergodicity
Ergodicity is the assumption that a system of certain temperature the whole phase space explored. In an Ising ferromagnet below that does not happen. The system selects a global rather magnetization, positive and negative values occur with equal probability, so that the phase-space is divided into two areas. It is not possible to get from one area to another, without applying a magnetic field or increasing the temperature above the critical temperature. In Heisenberg magnets even all arbitrary directions are below the critical temperature approved as equivalent - separate " Ergozitätskomponenten ", the description of the transition as a "critical phenomenon " (especially with the above critical exponents ) is still valid. In the so-called spin glasses - certain disordered spin systems - shall no longer, at least not in three dimensions, essentially because there they have a continuum of non - equivalent separate Ergozitätskomponenten.
Critical exponents and universality
For critical phenomenon is generally considered that the observables behave when approaching the critical point as an exponent. Here, the exponent is provided above and below, in general the same value. The exponent is negative in the case of divergence, convergence on the other hand, positive. If the value is logarithmic divergence or discontinuous behavior is possible. The exponents for different physical quantities are called critical exponents and are characteristic observables which are particularly sensitive to interference, provided they do not alter the symmetry of the system.
There are various scales relationships like between the critical exponents. Thereby call, and the critical exponent of the correlation function, the susceptibility and the correlation length. This phenomenon is referred to as "scaling ." In addition, " universal " applies, that is, while the above-mentioned exponent depends on the dimension of the system and of this symmetry, but have in each case an infinite large class of models have the same value. Both the "scaling " as well as the existence of universality classes can be explained qualitatively and quantitatively by the renormalization group theory.
Critical dynamics
Also in dynamic phenomena, there are critical behavior and Universality: The divergence of the characteristic time ( connected to other phenomena characteristic of the "critical deceleration " ) is recycled through a dynamic exponent on the divergence of the correlation length. The generally " very extensive " static University classes split into " less extensive " dynamic university classes, with different, but equally critical statics.
Critical opalescence
In certain liquid mixtures, there is a "critical opalescence " described phenomenon of " milky turbidity " are formed at the critical point of the liquid mixture more microscopic fine droplets, wherein the wavelength of the fluctuations but constantly increases the fluctuation dynamics simultaneously slows more and more
Mathematical Tools
Many features of the critical behavior can be derived from the renormalization group theory. These take advantage of the image of self-similarity to explain universality and predict numerical values of the critical exponents. It also depends on the variational perturbation theory, which changed divergent perturbation series in convergent developments of the strong coupling. The mean field theory is not suitable for the description of critical phenomena, as these are only far away from the phase transition is valid and neglected correlation effects, the gain in the vicinity of the critical point of importance because there is diverging correlation length.
In two-dimensional systems, the conformal field theory is a powerful tool. Taking advantage of scale invariance and a few other conditions that lead to infinite symmetry groups, a number of new properties of two-dimensional critical systems could be found.
Applications
Applications there are, in physics and chemistry also in subjects like sociology and Finance ( " Econophysics "). It is obvious, for example, to describe a two-party system ( näherungsweise! ) by an Ising model. In the transition from a majority opinion to the other you can then under certain circumstances. observe the critical phenomena described above.