![Wiener process animated](https://upload.wikimedia.org/wikipedia/commons/2/2a/Wiener_process_animated.gif)
In
physics,
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical term for this
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
is a dilatation (also known as dilation), and the dilatations can also form part of a larger
conformal symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilation (affine geometry), dilations. In three spatial plus one time dim ...
.
*In mathematics, scale invariance usually refers to an invariance of individual
functions or
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s. A closely related concept is
self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s of
random processes to display this kind of scale invariance or self-similarity.
*In
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
*In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, scale invariance has an interpretation in terms of
particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
*In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, scale invariance is a feature of
phase transitions. The key observation is that near a phase transition or
critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant
statistical field theories
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
, and are formally very similar to scale-invariant quantum field theories.
*
Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.
*In general,
dimensionless quantities are scale invariant. The analogous concept in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
are
standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.
Scale-invariant curves and self-similarity
In mathematics, one can consider the scaling properties of a
function or
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
under rescalings of the variable . That is, one is interested in the shape of for some scale factor , which can be taken to be a length or size rescaling. The requirement for to be invariant under all rescalings is usually taken to be
:
for some choice of exponent , and for all dilations . This is equivalent to being a
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
of degree .
Examples of scale-invariant functions are the
monomials
, for which , in that clearly
:
An example of a scale-invariant curve is the
logarithmic spiral, a kind of curve that often appears in nature. In
polar coordinates , the spiral can be written as
:
Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version of .
Projective geometry
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a
homogeneous polynomial, and more generally to a
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
. Homogeneous functions are the natural denizens of
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, and homogeneous polynomials are studied as
projective varieties in
projective geometry. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of
schemes, it has connections to various topics in
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.
Fractals
![Kochsim](https://upload.wikimedia.org/wikipedia/commons/6/65/Kochsim.gif)
It is sometimes said that
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s are scale-invariant, although more precisely, one should say that they are
self-similar
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
. A fractal is equal to itself typically for only a discrete set of values , and even then a translation and rotation may have to be applied to match the fractal up to itself.
Thus, for example, the
Koch curve scales with , but the scaling holds only for values of for integer . In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.
Some fractals may have multiple scaling factors at play at once; such scaling is studied with
multi-fractal analysis
A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed. ...
.
Periodic
external and internal rays are invariant curves .
Scale invariance in stochastic processes
If is the
average, expected power at frequency , then noise scales as
:
with = 0 for
white noise, = −1 for
pink noise, and = −2 for
Brownian noise
]
In science, Brownian noise, also known as Brown noise or red noise, is the type of signal noise produced by Brownian motion, hence its alternative name of random walk noise. The term "Brown noise" does not come from the color, but after ...
(and more generally,
Brownian motion).
More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. This likelihood is given by the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
Examples of scale-invariant distributions are the
Pareto distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actua ...
and the
Zipfian distribution.
Scale invariant Tweedie distributions
Tweedie distributions
In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the cla ...
are a special case of
exponential dispersion models, a class of statistical models used to describe error distributions for the
generalized linear model
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and b ...
and characterized by
closure under additive and reproductive convolution as well as under scale transformation.
These include a number of common distributions: the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
,
Poisson distribution and
gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
, as well as more unusual distributions like the compound Poisson-gamma distribution, positive
stable distributions, and extreme stable distributions.
Consequent to their inherent scale invariance Tweedie
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s ''Y'' demonstrate a
variance var(''Y'') to
mean E(''Y'') power law:
:
,
where ''a'' and ''p'' are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling,
and in the ecology literature as
Taylor's law
Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first propos ...
.
Random sequences, governed by the Tweedie distributions and evaluated by the
method of expanding bins exhibit a
biconditional relationship between the variance to mean power law and power law
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
s. The
Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest
''1/f'' noise.
The
Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.
It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a
variance function
In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statisti ...
that comes within the
domain of attraction of a Tweedie model. Almost all distribution functions with finite
cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this
asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.
Much as the
central limit theorem requires certain kinds of random variables to have as a focus of convergence the
Gaussian distribution and express
white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.
Cosmology
In
physical cosmology, the power spectrum of the spatial distribution of the
cosmic microwave background is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, , of
primordial fluctuations
Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe. Currently, the most widely accepted explanation for their origin is in the context of cosmic inflation. According ...
as a function of
wave number, , is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of
cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from seconds after the conjectured Big Bang singularity ...
.
Scale invariance in classical field theory
Classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
is generically described by a field, or set of fields, ''φ'', that depend on coordinates, ''x''. Valid field configurations are then determined by solving
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for ''φ'', and these equations are known as
field equation
In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equat ...
s.
For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,
:
:
The parameter ''Δ'' is known as the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
of the field, and its value depends on the theory under consideration. Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.
A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution, ''φ''(''x''), one always has other solutions of the form
:
.
Scale invariance of field configurations
For a particular field configuration, ''φ''(''x''), to be scale-invariant, we require that
:
where ''Δ'' is, again, the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
of the field.
We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will not be scale-invariant, and in such cases the symmetry is said to be
spontaneously broken
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
.
Classical electromagnetism
An example of a scale-invariant classical field theory is
electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,''t'') and B(x,''t''), while their field equations are
Maxwell's equations.
With no charges or currents,
these field equations take the form of
wave equations
:
:
where ''c'' is the speed of light.
These field equations are invariant under the transformation
:
:
Moreover, given solutions of Maxwell's equations, E(x, ''t'') and B(x, ''t''), it holds that
E(λx, λ''t'') and B(λx, λ''t'') are also solutions.
Massless scalar field theory
Another example of a scale-invariant classical field theory is the massless
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
(note that the name
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is unrelated to scale invariance). The scalar field, is a function of a set of spatial variables, ''x'', and a time variable, .
Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,
:
and is invariant under the transformation
:
:
The name massless refers to the absence of a term
in the field equation. Such a term is often referred to as a `mass' term, and would break the invariance under the above transformation. In
relativistic field theories, a mass-scale, is physically equivalent to a fixed length scale through
:
and so it should not be surprising that massive scalar field theory is ''not'' scale-invariant.
φ4 theory
The field equations in the examples above are all
linear in the fields, which has meant that the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, , has not been so important. However, one usually requires that the scalar field
action is dimensionless, and this fixes the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
of . In particular,
:
where is the combined number of spatial and time dimensions.
Given this scaling dimension for , there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless
φ4 theory for =4. The field equation is
:
(Note that the name
4 derives from the form of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, which contains the fourth power of .)
When =4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is =1. The field equation is then invariant under the transformation
:
:
:
The key point is that the parameter must be dimensionless, otherwise one introduces a fixed length scale into the theory: For
4 theory, this is only the case in =4.
Note that under these transformations the argument of the function is unchanged.
Scale invariance in quantum field theory
The scale-dependence of a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
(QFT) is characterised by the way its
coupling parameters depend on the energy-scale of a given physical process. This energy dependence is described by the
renormalization group, and is encoded in the
beta-functions of the theory.
For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as
fixed points of the corresponding renormalization group flow.
Quantum electrodynamics
A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since
photons are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.
However, in nature the electromagnetic field is coupled to charged particles, such as
electrons. The QFT describing the interactions of photons and charged particles is
quantum electrodynamics (QED), and this theory is not scale-invariant. We can see this from the
QED beta-function. This tells us that the
electric charge (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles is scale-invariant, QED is not scale-invariant.
Massless scalar field theory
Free, massless
quantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the
Gaussian fixed point
A Gaussian fixed point is a fixed point of the renormalization group flow which is noninteracting in the sense that it is described by a free field theory. The word Gaussian comes from the fact that the probability distribution is Gaussian at the ...
.
However, even though the classical massless ''φ''
4 theory is scale-invariant in ''D''=4, the quantized version is not scale-invariant. We can see this from the
beta-function for the coupling parameter, ''g''.
Even though the quantized massless ''φ''
4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson–Fisher fixed point, below.
Conformal field theory
Scale-invariant QFTs are almost always invariant under the full
conformal symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilation (affine geometry), dilations. In three spatial plus one time dim ...
, and the study of such QFTs is
conformal field theory (CFT).
Operators in a CFT have a well-defined
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, analogous to the
scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
, ''∆'', of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as
anomalous scaling dimension
In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime dilations x\to \lambda x. If the quantum field theory is sca ...
s.
Scale and conformal anomalies
The φ
4 theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be
anomalous. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called
cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from seconds after the conjectured Big Bang singularity ...
, as long as the theory can be studied through
perturbation theory.
Phase transitions
In
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, as a system undergoes a
phase transition, its fluctuations are described by a scale-invariant
statistical field theory. For a system in equilibrium (i.e. time-independent) in spatial dimensions, the corresponding statistical field theory is formally similar to a -dimensional CFT. The scaling dimensions in such problems are usually referred to as
critical exponent
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 in ...
s, and one can in principle compute these exponents in the appropriate CFT.
The Ising model
An example that links together many of the ideas in this article is the phase transition of the
Ising model, a simple model of
ferromagnetic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a -dimensional periodic lattice. Associated with each lattice site is a
magnetic moment, or
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
, and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)
The key point is that the Ising model has a spin-spin interaction, making it energetically favourable for two adjacent spins to be aligned. On the other hand, thermal fluctuations typically introduce a randomness into the alignment of spins. At some critical temperature, ,
spontaneous magnetization
Spontaneous magnetization is the appearance of an ordered spin state (magnetization) at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or .
Overview
Heated to temperat ...
is said to occur. This means that below the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.
An example of the kind of physical quantities one would like to calculate at this critical temperature is the correlation between spins separated by a distance . This has the generic behaviour:
:
for some particular value of
, which is an example of a critical exponent.
CFT description
The fluctuations at temperature are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the Wilson–Fisher fixed point, a particular scale-invariant
scalar field theory
In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.
The only fundamental scalar quantum field that has b ...
.
In this context, is understood as a
correlation function
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 rep ...
of scalar fields,
:
Now we can fit together a number of the ideas seen already.
From the above, one sees that the critical exponent, , for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field,
:
is modified to become
:
where is the number of dimensions of the Ising model lattice.
So this anomalous dimension in the conformal field theory is the ''same'' as a particular critical exponent of the Ising model phase transition.
Note that for dimension , can be calculated approximately, using the epsilon expansion, and one finds that
:
.
In the physically interesting case of three spatial dimensions, we have =1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that is numerically small in three dimensions.
On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the
minimal models, a family of well-understood CFTs, and it is possible to compute (and the other critical exponents) exactly,
:
.
Schramm–Loewner evolution
The anomalous dimensions in certain two-dimensional CFTs can be related to the typical
fractal dimensions of random walks, where the random walks are defined via
Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensiona ...
(SLE). As we have seen above, CFTs describe the physics of phase transitions, and so one can relate the critical exponents of certain phase transitions to these fractal dimensions. Examples include the 2''d'' critical Ising model and the more general 2''d'' critical
Potts model. Relating other 2''d'' CFTs to SLE is an active area of research.
Universality
A phenomenon known as
universality is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:
* The
Ising model phase transition, described above.
* The
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
-
vapour transition in classical fluids.
Even though the microscopic physics of these two systems is completely different, their critical exponents turn out to be the same. Moreover, one can calculate these exponents using the same statistical field theory. The key observation is that at a phase transition or
critical point, fluctuations occur at all length scales, and thus one should look for a scale-invariant statistical field theory to describe the phenomena. In a sense, universality is the observation that there are relatively few such scale-invariant theories.
The set of different microscopic theories described by the same scale-invariant theory is known as a
universality class
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
. Other examples of systems which belong to a universality class are:
*
Avalanches in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales.
* The frequency of
network outages on the
Internet, as a function of size and duration.
* The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper.
* The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
* The
electrical breakdown of
dielectrics, which resemble cracks and tears.
* The
percolation of fluids through disordered media, such as
petroleum through fractured rock beds, or water through filter paper, such as in
chromatography. Power-law scaling connects the rate of flow to the distribution of fractures.
* The
diffusion of
molecules in
solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
, and the phenomenon of
diffusion-limited aggregation
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is app ...
.
* The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
The key observation is that, for all of these different systems, the behaviour resembles a
phase transition, and that the language of statistical mechanics and scale-invariant
statistical field theory may be applied to describe them.
Other examples of scale invariance
Newtonian fluid mechanics with no applied forces
Under certain circumstances,
fluid mechanics is a scale-invariant classical field theory. The fields are the velocity of the fluid flow,
, the fluid density,
, and the fluid pressure,
. These fields must satisfy both the
Navier–Stokes equation and the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
. For a
Newtonian fluid these take the respective forms
:
where
is the
dynamic 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 ...
.
In order to deduce the scale invariance of these equations we specify an
equation of state, relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the
isothermal ideal gas, which satisfies
:
where
is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations
:
:
:
:
Given the solutions
and
, we automatically have that
and
are also solutions.
Hidden scale invariance in liquids and solids
Certain models studied by
Molecular Dynamics computer simulations, including the
Lennard-Jones
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical science at the University of Cambridge. He was an imp ...
and
Yukawa Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns.
People
* Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1
* Hideki Yukawa (1907–1 ...
pair-potential models, have a potential-energy function
that to a good approximation obeys the "hidden scale invariance" criterion