In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, critical phenomena is the collective name associated with the
physics of
critical points. Most of them stem from the divergence of the
correlation length
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables re ...
, but also the dynamics slows down. Critical phenomena include
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
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
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
breaking. Critical phenomena take place in
second order phase transitions, although not exclusively.
The critical behavior is usually different from the
mean-field approximation
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over Degrees of ...
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 theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
.
In order to explain the physical origin of these phenomena, we shall use the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
as a pedagogical example.
The critical point of the 2D Ising model
Consider a
square array of classical spins which may only take two positions: +1 and −1, at a certain temperature
, interacting through the
Ising classical
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
:
:
where the sum is extended over the pairs of nearest neighbours and
is a coupling constant, which we will consider to be fixed. There is a certain temperature, called the
Curie temperature
In physics and materials science, the Curie temperature (''T''C), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Cur ...
or
critical temperature,
below which the system presents
ferromagnetic long range order. Above it, it is
paramagnetic
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...
and is apparently disordered.
At temperature zero, the system may only take one global sign, either +1 or -1. At higher temperatures, but below
, the state is still globally magnetized, but clusters of the opposite sign appear. 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
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables re ...
,
grows with temperature until it diverges at
. 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
,
. 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
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...
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
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
, 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
for some exponent
where, typically, the value of the exponent α is the same above and below T
c. These exponents are called
critical exponents
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
* Critical juncture, a discontinuous change studied in the social sciences.
* Critical Software, a company specializing i ...
and are robust observables. Even more, they take the same values for very different physical systems. This intriguing phenomenon, called
universality, is explained, qualitatively and also quantitatively, by the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
.
Critical dynamics
Critical phenomena may also appear for ''dynamic'' quantities, not only for ''static'' ones. In fact, the divergence of the characteristic ''time''
of a system is directly related to the divergence of the thermal ''correlation length''
by the introduction of a dynamical exponent ''z'' and the relation
. 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. The divergence of relaxation time
at criticality leads to singularities in various collective transport quantities, e.g., the interdiffusivity,
shear viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
, and bulk viscosity
. The dynamic critical exponents follow certain scaling relations, viz.,
, where d is the space dimension. There is only one independent dynamic critical exponent. Values of these exponents are dictated by several universality classes. According to the Hohenberg−Halperin nomenclature, for the model H universality class (fluids)
.
Ergodicity breaking
Ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
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
this does not happen. If