Scale invariance

In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry. In Mathematics a transform is an Operator applied to a function so that under the transform certain operations are simplified In theoretical physics conformal symmetry is a Symmetry under dilatation ( Scale invariance) and under the special conformal transformations.

In mathematics, scale invariance usually refers to an invariance of individual functions or curves. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilatations. In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory.

A Koch curve is scale-invariant.

In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. A classical field theory is a Physical theory that describes the study of how one or more physical fields interact with matter Such theories typically describe classical physical processes with no characteristic length scale.

In quantum field theory, scale invariance has an interpretation in terms of particle physics. In quantum field theory (QFT the forces between particles are mediated by other particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.

In statistical mechanics, scale invariance is a feature of phase transitions. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another 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, and are formally very similar to scale-invariant quantum field theories. A statistical field theory is any model in Statistical mechanics where the degrees of freedom comprise a field or fields

Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. 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 Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.

In general, dimensionless quantities are scale invariant. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units The analogous concept in statistics are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, the k th standardized moment of a Probability distribution is \frac{\mu_k}{\sigma^k}\! where

Scale-invariant curves and self-similarity

In mathematics, one can consider the scaling properties of a function or curve $f (x)$ under rescalings of the variable $x$ . The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object That is, one is interested in the shape of $f (λ x)$ for some scale factor $λ$ , which can be taken to be a length or size rescaling. The requirement for $f (x)$ to be invariant under all rescalings is usually taken to be

f (x) = λ - Δ f (λ x)

for some choice of exponent $Δ$ , and for all dilations $λ$ .

Examples of scale-invariant functions are the monomials $f (x) = x n$ , for which one has $Δ = n$ , in that clearly

f (λ x) = (λ x) n = λ n f (x). In Mathematics, the word monomial means two different things in the context of Polynomials The first meaning is a product of powers of Variables

An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta In polar coordinates (r, θ) the spiral can be written as

$\theta = \frac{1}{b} \ln(r/a).$

Allowing for rotations of the curve, it is invariant under all rescalings $λ$ ; that is $θ(λ r)$ is identical to a rotated version of $θ(r)$ . In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

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 polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. 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 Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

Fractals

It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. 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" In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i A fractal is equal to itself typically for only a discrete set of values $λ$ , and even then a translation and rotation must be applied to match up to the fractal to itself. Thus, for example the Koch curve scales with $Δ = 1$ , but the scaling holds only for values of $λ = 1 / 3 n$ for integer n. The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described 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

Scale invariance in stochastic processes

If $P (f)$ is the average, expected power at frequency $f$ , then noise scales as

P (f) = λ - Δ P (λ f)

with $Δ = 0$ for white noise, $Δ = - 1$ for pink noise, and $Δ = - 2$ for Brownian noise (and more generally, Brownian motion). Pink noise or 1/f noise is a signal or process with a Frequency spectrum such that the power spectral density is Proportional In Science, Brownian noise ( also known as Brown noise or red noise, is the kind of Signal noise produced by Brownian motion This article is about the physical phenomenon for the stochastic process see Wiener process.

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 identifies either the probability of each value of an unidentified Random variable Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution. The Pareto distribution, named after the Italian Economist Vilfredo Pareto, is a Power law Probability distribution that coincides with WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one

Cosmology

In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its This means that the amplitude, $P (k)$ , of primordial fluctuations as a function of wave number, $k$ , is approximately constant. Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters This pattern is consistent with the proposal of cosmic inflation. In Physical cosmology, cosmic inflation is the idea that the nascent Universe passed through a phase of exponential expansion that

Scale invariance in classical field theory

Classical field theory is generically described by a field, or set of fields, $\varphi$ , that depend on coordinates, $x$ . A classical field theory is a Physical theory that describes the study of how one or more physical fields interact with matter Valid field configurations are then determined by solving differential equations for $\varphi(x)$ , and these equations are known as field equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the A field equation is an equation in a Physical theory that describes how a Fundamental force (or a combination of such forces interacts with Matter

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:

$x\rightarrow\lambda x,$

$\varphi\rightarrow\lambda^{-\Delta}\varphi.$

The parameter $Δ$ is known as the scaling dimension 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, $\varphi(x)$ , one always has other solutions of the form $\lambda^{-\Delta}\varphi(\lambda x)$ .

Scale invariance of field configurations

For a particular field configuration, $\varphi(x)$ , to be scale-invariant, we require that

$\varphi(x)=\lambda^{-\Delta}\varphi(\lambda x)$

where $Δ$ is again the scaling dimension 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. In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state

Classical electromagnetism

An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The electromagnetic field is a physical field produced by electrically charged objects. The fields are the electric and magnetic fields, $\mathbf{E}(\mathbf{x},t)$ and $\mathbf{B}(\mathbf{x},t)$ , while their field equations are Maxwell's equations. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric With no charges or currents, these field equations take the form of wave equations

$\nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}$

$\nabla^2\mathbf{B} = \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2}$

where c is the speed of light. The electromagnetic field is a physical field produced by electrically charged objects. The wave equation is an important second-order linear Partial differential equation that describes the propagation of a variety of Waves such as Sound waves

These field equations are invariant under the transformation

$x\rightarrow\lambda x,$

$t\rightarrow\lambda t.$

Moreover, given solutions of Maxwell's equations, $\mathbf{E}(\mathbf{x},t)$ and $\mathbf{B}(\mathbf{x},t)$ , we have that $\mathbf{E}(\lambda\mathbf{x},\lambda t)$ and $\mathbf{B}(\lambda\mathbf{x},\lambda t)$ are also solutions.

Massless scalar field theory

Another example of a scale-invariant classical field theory is the massless scalar field (note that the name scalar is unrelated to scale invariance). For the pseudoscientific "scalar field theory" see " Scalar field theory (pseudoscience " In Theoretical physics, The scalar field, $\varphi(\mathbf{x},t)$ is a function of a set of spatial variables, $\mathbf{x}$ , and a time variable, t. We first consider the linear theory. Much like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation

$\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi = 0,$

and is invariant under the transformation

$x\rightarrow\lambda x,$

$t\rightarrow\lambda t.$

The name massless refers to the absence of a term $\propto m^2\varphi^2$ 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, $m$ is physically equivalent to a fixed length scale via

$L=\frac{\hbar}{mc},$

and so it should not be surprising that massive scalar field theory is not scale-invariant. In Physics, a field is a Physical quantity associated to each point of Spacetime.

φ⁴ theory

The field equations in the examples above are all linear in the fields, which has meant that the scaling dimension, $Δ$ , has not been so important. The word linear comes from the Latin word linearis, which means created by lines. However, one usually requires that the scalar field action is dimensionless, and this fixes the scaling dimension of $\varphi$ . In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In particular,

$\Delta=\frac{D-2}{2},$

where D is the combined number of spatial and time dimensions.

Given this scaling dimension for $\varphi$ , there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless φ⁴ theory for $D = 4$ . In Quantum field theory, a quartic interaction is a theory about a Scalar field &phi which contains an interaction term \phi^4 and is considered by The field equation is

$\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.$

(Note that the name $\varphi^4$ derives from the form of the Lagrangian, which contains the fourth power of $\varphi$ . In Quantum field theory, a quartic interaction is a theory about a Scalar field &phi which contains an interaction term \phi^4 and is considered by )

When D=4 (e. g. three spatial dimensions and one time dimension), the scalar field scaling dimension is $Δ = 1$ . The field equation is then invariant under the transformation

$x\rightarrow\lambda x,$

$t\rightarrow\lambda t,$

$\varphi\rightarrow\lambda^{-1}\varphi.$

The key point is that the parameter g must be dimensionless, otherwise one introduces a fixed length scale into the theory. For φ⁴ theory this is only the case in $D = 4$ .

Scale invariance in quantum field theory

The scale-dependence of a quantum field theory (QFT) is characterised by the way its coupling parameters depend on the energy-scale of a given physical process. In quantum field theory (QFT the forces between particles are mediated by other particles In Physics, a coupling constant, usually denoted g, is a number that determines the strength of an Interaction. This energy dependence is described by the renormalization group, and is encoded in the beta-functions of the theory. 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 Theoretical physics, specifically Quantum field theory, a beta-function β(g encodes the dependence of a coupling parameter, g on the energy scale

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. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

However, in nature the electromagnetic field is coupled to charged particles, such as electrons. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The QFT describing the interactions of photons and charged particles is quantum electrodynamics (QED), and this theory is not scale-invariant. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. We can see this from the QED beta-function. In Theoretical physics, specifically Quantum field theory, a beta-function β(g encodes the dependence of a coupling parameter, g on the energy scale This tells us that the electric charge (which is the coupling parameter in the theory) increases with increasing energy. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. 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. In Quantum field theory, a bosonic field is a Quantum field whose quanta are Bosons that is they obey Bose-Einstein statistics. 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

However, even though the classical massless φ⁴ 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. In Theoretical physics, specifically Quantum field theory, a beta-function β(g encodes the dependence of a coupling parameter, g on the energy scale

Even though the quantized massless φ⁴ 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.

Conformal field theory

Scale-invariant QFTs are almost always invariant under the full conformal symmetry, and the study of such QFTs is conformal field theory (CFT). In theoretical physics conformal symmetry is a Symmetry under dilatation ( Scale invariance) and under the special conformal transformations. A conformal field theory (CFT is a Quantum field theory (or Statistical mechanics model at the Critical point) that is Invariant under Operators in a CFT have a well-defined scaling dimension, analogous to the scaling dimension, $Δ$ , of a classical field discussed above. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained However, the scaling dimensions of operators in a CFT typically differ from the those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as anomalous scaling dimensions. In Theoretical physics, by anomaly one usually means that the symmetry remains broken when the symmetry-breaking factor goes to zero

Scale and conformal anomalies

The φ⁴ 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. Conformal anomaly is an anomaly ie a quantum phenomenon that breaks the Conformal symmetry of the Classical theory.

Phase transitions

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another A statistical field theory is any model in Statistical mechanics where the degrees of freedom comprise a field or fields For a system in equilibrium (i. e. time-independent) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D-dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT. Critical exponents describe the behaviour of physical quantities near continuous Phase transitions.

The Ising model

An example that links together many of the ideas in this article is the phase transition of the Ising model, a crude model of ferromagnetic substances. The Ising model, named after the physicist Ernst Ising, is a mathematical model in Statistical mechanics. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it 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 D-dimensional periodic lattice. Associated with each lattice site is a magnetic moment, or spin, and this spin can take either the value +1 or -1. In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin (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, $T c$ , spontaneous magnetization is said to occur. Spontaneous magnetization is the term used to describe the appearance of an ordered spin state ( Magnetization) at zero applied magnetic field in a Ferromagnetic This means that below $T c$ 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 r. This has the generic behaviour:

$G(r)\propto\frac{1}{r^{D-2+\eta}},$

for some particular value of $η$ , which is an example of a critical exponent.

CFT description

The fluctuations at temperature $T c$ 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 Quantum field theory, a bosonic field is a Quantum field whose quanta are Bosons that is they obey Bose-Einstein statistics. In this context, $G (r)$ is understood as a correlation function of scalar fields:

$\langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}.$

Now we can fit together a number of the ideas we've seen already. Correlation functions contain information about the distribution of points or events or things across some space/time From the above we can see that the critical exponent, $η$ , for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field

$\Delta=\frac{D-2}{2}$

is modified to become

$\Delta=\frac{D-2+\eta}{2},$

where D 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.

We note that for dimension $D = 4 - ε$ , $η$ can be calculated approximately, using the epsilon expansion, and one finds that

$\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)$ .

In the physically interesting case of three spatial dimensions we have $ε = 1$ , and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that $η$ 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 $η$ (and the other critical exponents) exactly:

$\eta_{D=2}=\frac{1}{4}$ . In theoretical physics the term minimal model usually refers to a special class of conformal field theories that generalize the Ising model or to some closely related

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 (SLE). In Fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a Fractal appears to fill space as 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 2d critical Ising model and the more general 2d critical Potts model. In Statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a Crystalline lattice. Relating other 2d CFTs to SLE is an active area of research.

Universality

A phenomenon known as universality is seen in a large variety of physical systems. 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 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 Ising model, named after the physicist Ernst Ising, is a mathematical model in Statistical mechanics.
The liquid-vapour transition in classical fluids. Liquid is one of the principal States of matter. A liquid is a Fluid that has the particles loose and can freely form a distinct surface at the boundaries of A vapor or vapour (see Spelling differences) is a substance in the Gas phase at a Temperature lower than its Critical temperature

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 Theoretical physics, renormalization group (RG refers to a mathematical apparatus that allows one to investigate the changes of a physical system as one views Other examples of systems which belong to a universality class are:

Avalanches in piles of sand. This article refers to the natural event For other uses see Avalanche (disambiguation An avalanche is an abrupt and rapid flow of snow often 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. Downtime or outage refers to a period of time or a percentage of a timespan that a System is unavailable or Offline. The Internet is a global system of interconnected Computer networks
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 term electrical breakdown has several similar but distinctly different meanings A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday.
The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. In Physics, Chemistry and Materials science, percolation concerns also the movement and filtering of fluids through porous materials Petroleum ( L petroleum, from Greek πετρέλαιον, lit Chromatography (from Greek χρώμα chroma, color and γραφειν"graphein" to write is the collective term for a family of Laboratory Power-law scaling connects the rate of flow to the distribution of fractures.
The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation. Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by In Chemistry, a solution is a Homogeneous Mixture composed of two or more substances 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
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. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another A statistical field theory is any model in Statistical mechanics where the degrees of freedom comprise a field or fields

Other examples of scale invariance

Newtonian fluid mechanics with no applied forces

Under certain circumstances, fluid mechanics is a scale-invariant classical field theory. Fluid mechanics is the study of how Fluids move and the Forces on them The fields are the velocity of the fluid flow, $\mathbf{u}(\mathbf{x},t)$ , the fluid density, $\rho(\mathbf{x},t)$ , and the fluid pressure, $P(\mathbf{x},t)$ . These fields must satisfy both the Navier–Stokes equation and the continuity equation. The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity For a Newtonian fluid these take the respective forms

$\rho\frac{\partial \mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\nabla \mathbf{u} = -\nabla P+\mu \left(\nabla^2 \mathbf{u}+\frac{1}{3}\nabla\left(\nabla\cdot\mathbf{u}\right)\right)$

$\frac{\partial \rho}{\partial t}+\nabla\cdot \left(\rho\mathbf{u}\right)=0$

where $μ$ is the dynamic viscosity. A Newtonian fluid (named for Isaac Newton) is a Fluid whose stress versus rate of strain curve is linear and passes through the origin Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.

In order to deduce the scale invariance of these equations we specify an equation of state, relating the fluid pressure to the fluid density. In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 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

$P=c_s^2\rho,$

where $c s$ is the speed of sound in the fluid. An isothermal process is a Thermodynamic process in which the Temperature of the System stays Constant: &Delta T = 0 These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations

$x\rightarrow\lambda x,$

$t\rightarrow\lambda t,$

$\rho\rightarrow\lambda \rho,$

$\mathbf{u}\rightarrow\mathbf{u}.$

Given the solutions $\mathbf{u}(\mathbf{x},t)$ and $\rho(\mathbf{x},t)$ , we automatically have that $\mathbf{u}(\lambda\mathbf{x},\lambda t)$ and $\lambda\rho(\lambda\mathbf{x},\lambda t)$ are also solutions.

Computer vision

In computer vision, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. Computer vision is the science and technology of machines that see A general framework for obtaining scale invariance in practice is by detecting local maxima over scales of normalized derivative responses -- see the article on scale-space for a brief introduction to the general theory and references. Scale-space theory is a framework for multi-scale signal representation developed by the Computer vision, Image processing and Examples of scale invariant blob detectors and ridge detectors are given in the articles on blob detection and ridge detection. In the area of Computer vision, ' blob detection' refers to visual modules that are aimed at detecting points and/or regions in the image that are either brighter or darker The ridges (or the ridge set) of a smooth function of two variables is a set of curves whose points are loosely speaking local maxima in at least one dimension An example of the application of scale invariance to object recognition is given in the article on the scale-invariant feature transform. Scale-invariant feature transform (or SIFT) is an algorithm in Computer vision to detect and describe local features in images

References

Zinn-Justin, Jean ; Quantum Field Theory and Critical Phenomena, Oxford University Press (2002). Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.

Dictionary

-noun

(physics) (mathematics) a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor.

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Contents

Scale-invariant curves and self-similarity

Projective geometry

Fractals

Scale invariance in stochastic processes

Cosmology

Scale invariance in classical field theory

Scale invariance of field configurations

Classical electromagnetism

Massless scalar field theory

φ4 theory

Scale invariance in quantum field theory

Quantum electrodynamics

Massless scalar field theory

Conformal field theory

Scale and conformal anomalies

Phase transitions

The Ising model

CFT description

Schramm-Loewner evolution

Universality

Other examples of scale invariance

Newtonian fluid mechanics with no applied forces

Computer vision

References

Dictionary

scale invariance

-noun

φ⁴ theory