Critical phenomena - Citizendia

In physics, critical phenomena is the collective name associated with the physics of critical points. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Physical chemistry, Thermodynamics, Chemistry and Condensed matter physics, a critical point, also called a critical state Most of them stem from the divergence of the correlation length. The Correlation function in Statistical mechanics is measure of the order in a system 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. A power law is any Polynomial relationship that exhibits the property of Scale invariance. In Electromagnetism the magnetic susceptibility ( Latin: susceptibilis “receptiveness” is the degree of Magnetization of a material in response Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it Critical exponents describe the behaviour of physical quantities near continuous 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 A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems Critical phenomena take place in second order phase transition, although not exclusively. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another

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. A Many-body system with interactions is generally very difficult to solve exactly except for extremely simple cases ( Gaussian field theory, 1D Ising model. Many properties of the critical behavior of a system can be derived in the framework of the renormalization group. In Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views

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.

1 The critical point of the 2D Ising model
2 Divergences at the critical point
3 Critical exponents and universality
4 Critical dynamics
5 Ergodicity breaking
6 Mathematical tools
7 See also
8 Bibliography

The critical point of the 2D Ising model

Let us consider a $2 D$ 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:

$H= -J \sum_{[i,j]} S_i\cdot S_j$

where the sum is extended over the pairs of nearest neighbours and $J$ is a coupling constant, which we will consider to be fixed. Ernst Ising (born May 10, 1900, Cologne, Germany &ndash May 11, 1998, Peoria, Illinois, There is a certain temperature, called the Curie temperature or critical temperature, $T c$ below which the system presents ferromagnetic long range order. The Curie point ( Tc) or Curie temperature, is a term in Physics and Materials science, named after Pierre Curie (1859-1906 The critical temperature, Tc of a material is the Temperature above which distinct Liquid and Gas phases do not exist Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it Above it, it is paramagnetic and is apparently disordered. Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied 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 $T c$ . The Correlation function in Statistical mechanics is measure of the order in a system 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.

Divergences at the critical point

The correlation length diverges at the critical point: as $T\to T_c$ , $\xi\to\infty$ . 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. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" It affects easily the smallest size clusters, since they have a nearly paramagnetic behaviour. Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field 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. Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity 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 $A(T)\approx (T-T_c)^\alpha$ for some exponent $α$ . These exponents are called critical exponents and are robust observables. Critical exponents describe the behaviour of physical quantities near continuous Phase transitions. Even more, they take the same values for very different physical systems. This intriguing phenomenon, called universality, is explained successfully by the renormalization group. In Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views

Critical dynamics

Critical phenomena may also appear for dynamic quantities, not only for static ones. In fact, the divergence of the characteristic time $τ$ of a system is directly related to the divergence of the thermal correlation length $ξ$ by the introduction of a dynamical exponent z and the relation $\tau =\xi^{\,z}$ . 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.

Ergodicity breaking

Ergodicity is the assumption that a system, at a given temperature, explores the full phase space, just each state takes different probabilities. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems 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 $T c$ .

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. A superselection sector is a concept used in Quantum mechanics when a representation of a *-algebra is decomposed into irreducible components In Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views In Mathematics, variational perturbation theory is a mathematical method to convert divergent Power series in a small expansion parameter say s=\sum_{n=0}^\infty 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. A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is

Bibliography

Phase Transitions and Critical Phenomena, vol. The Ising model, named after the physicist Ernst Ising, is a mathematical model in Statistical mechanics. Critical exponents describe the behaviour of physical quantities near continuous Phase transitions. Critical opalescence is a phenomenon which arises in the region of a continuous or second-order Phase transition. In Mathematics, variational perturbation theory is a mathematical method to convert divergent Power series in a small expansion parameter say s=\sum_{n=0}^\infty A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems In Physics, self-organized criticality (SOC is a property of (classes of Dynamical systems which have a critical point as an Attractor. In Statistical mechanics, the Rushbrooke inequality relates the Critical exponents of a Magnetic system which exhibits a first-order Widom scaling is a hypothesis in Statistical mechanics regarding the free energy of a Magnetic system near its Critical point which leads to 1-20 (1972-2001), Academic Press, Ed. : C. Domb, M. Cyril Domb (born 9 December 1920) is a British scientist now living in Israel. S. Green, J.L. Lebowitz
J. Joel L Lebowitz (born May 10, 1930 in Taceva) is a Jewish Mathematical physicist widely acknowledged for his outstanding contributions 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. Hagen Kleinert (born 1941 is Professor of Theoretical Physics at the Free University of Berlin, Germany (since 1968 Honorary Professor at the Kyrgyz-Russian Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4659-5 (Read online at [1])
J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford Science Publications, 1992) ISBN 0198517300
M.E. Fisher, Renormalization Group in Theory of Critical Behavior, Reviews of Modern Physics, vol. Michael Ellis Fisher (born 3 September 1931 in Trinidad) is a physicist as well as chemist and mathematician known for his many seminal contributionsto 46, p. 597-616 (1974)

Contents