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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. 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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In mathematics, a transformation in elementary terms is any of a variety of different functions from geometry, such as rotations, reflections and translations. ...
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space...
- In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilatations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
 A Koch curve is scale-invariant. - In classical field theory, 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, 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, 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, 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 are standardized moments, which are scale invariant statistics of a variable, while the unstandardized moments are not.
This article is about functions in mathematics. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
A self-similar object is exactly or approximately similar to a part of itself. ...
A probability distribution describes the values and probabilities that a random event can take place. ...
In the mathematics of probability, a stochastic process can be thought of as a random function. ...
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. ...
Quantum field theory (QFT) is the quantum theory of fields. ...
Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...
Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
This diagram shows the nomenclature for the different phase transitions. ...
The term critical point can mean any of: critical point (thermodynamics) critical point (mathematics) critical loops (topology) critical point (set theory) This is a disambiguation page: a list of articles associated with the same title. ...
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. ...
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 system. ...
This article is about the field of statistics. ...
In probability theory and statistics, the kth standardized moment of a probability distribution is μk/σk, where μk is the kth moment about the mean and σ is the standard deviation. ...
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. That is, one is interested in the shape of f(λx) 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 This article is about functions in mathematics. ...
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
- f(x) = λ − Δf(λx)
for some choice of exponent Δ, and for all dilations λ.
Examples of scale-invariant functions are the monomials f(x) = xn, for which one has Δ = n, in that clearly In mathematics, a monomial (or mononomial) is a particular kind of polynomial, having just one term. ...
- f(λx) = (λx)n = λnf(x).
An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (r, θ) the spiral can be written as A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ...
This article describes some of the common coordinate systems that appear in elementary mathematics. ...
 Allowing for rotations of the curve, it is invariant under all rescalings λ; that is θ(λr) is identical to a rotated version of θ(r).
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. Homogeneous functions are the natural denizens of projective space, 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 mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. ...
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...
This article does not cite its references or sources. ...
This article is about algebraic varieties. ...
Projective geometry is a non-metrical form of geometry. ...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
This box: String theory is a still developing mathematical 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 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 / 3n for integer n. 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. The boundary of the Mandelbrot set is a famous example of a fractal. ...
A self-similar object is exactly or approximately similar to a part of itself. ...
The first four iterations of the Koch snowflake. ...
Some fractals may have multiple scaling factors at play at once; such scaling is studied with multi-fractal analysis.
Scale invariance in stochastic processes If P(f) is the average, expected power at frequency f, then noise scales as In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are...
- P(f) = λ − ΔP(λf)
with Δ = 0 for white noise, Δ = − 1 for pink noise, and Δ = − 2 for Brownian noise (and more generally, Brownian motion). Pink noise spectrum Pink noise ( ), also known as 1/f noise or flicker noise, is a signal or process with a frequency spectrum such that the power spectral density is proportional to the reciprocal of the frequency. ...
Brown noise spectrum In science, Brownian noise ( ), also known as Brown noise or red noise, is the kind of signal noise produced by Brownian motion. ...
Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...
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. Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution. A probability distribution describes the values and probabilities that a random event can take place. ...
The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...
Originally the term Zipfs law meant the observation of Harvard linguist George Kingsley Zipf (SAMPA: [zIf]) that the frequency of use of the nth-most-frequently-used word in any natural language is approximately inversely proportional to n. ...
Cosmology In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function. This means that the amplitude, P(k), of primordial fluctuations as a function of wave number, k, is approximately constant. This pattern is consistent with the proposal of cosmic inflation. This article is about the physics subject. ...
WMAP image of the CMB anisotropy,Cosmic microwave background radiation(June 2003) The cosmic microwave background radiation (CMB) is a form of electromagnetic radiation that fills the whole of the universe. ...
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 units of inverse length. ...
In physical cosmology, cosmic inflation is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density. ...
Scale invariance in classical field theory Classical field theory is generically described by a field, or set of fields,  , that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for  , and these equations are known as field equations. A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. ...
In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
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:
  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, , one always has other solutions of the form  .
Scale invariance of field configurations For a particular field configuration, , to be scale-invariant, we require that  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. Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ...
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,  and  , while their field equations are Maxwell's equations. With no charges or currents, these field equations take the form of wave equations The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...
For thermodynamic relations, see Maxwell relations. ...
The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ...
The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...
  where c is the velocity of light.
These field equations are invariant under the transformation
 Moreover, given solutions of Maxwell's equations, and , we have that  and  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). The scalar field,  is a function of a set of spatial variables,  , 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 It has been suggested that quartic interaction be merged into this article or section. ...
A scalar may be: Look up scalar in Wiktionary, the free dictionary. ...
 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, m is physically equivalent to a fixed length scale via  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, Δ, has not been so important. However, one usually requires that the scalar field action is dimensionless, and this fixes the scaling dimension of . In particular, For other uses, see Linear (disambiguation). ...
In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...
 where D 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 D = 4. The field equation is This article is in need of attention from an expert on the subject. ...
 (Note that the name  derives from the form of the Lagrangian, which contains the fourth power of .) This article is in need of attention from an expert on the subject. ...
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
  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. This energy dependence is described by the renormalization group, and is encoded in the beta-functions of the theory. Quantum field theory (QFT) is the quantum theory of fields. ...
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. ...
In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...
In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by for Re(x), Re(y) > 0. ...
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. In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...
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 modern 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 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. For other uses, see Electron (disambiguation). ...
Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...
In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by for Re(x), Re(y) > 0. ...
This box: Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ...
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. It has been suggested that this article or section be merged into Scalar field theory. ...
See UV fixed point, IR 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 mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by for Re(x), Re(y) > 0. ...
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). Operators in a CFT have a well-defined scaling dimension, analogous to the scaling dimension, Δ, of a classical field discussed above. 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 mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space...
A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ...
In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...
In theoretical physics, namely quantum field theory, the anomalous scaling dimension of an operator is the contribution of quantum mechanics to the classical scaling dimension of that operator. ...
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 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. 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. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
This diagram shows the nomenclature for the different phase transitions. ...
A statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field or fields. ...
The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...
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. 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. (These states are also called up and down, respectively.) The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ...
A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ...
A bar magnet. ...
In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ...
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, Tc, spontaneous magnetization is said to occur. This means that below Tc the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions. Spontaneous magnetization is the term used to describe the appearance of an ordered spin state at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or TC. At temperatures above TC, the material is paramagnetic and its magnetic behavior is dominated...
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:
 for some particular value of η, which is an example of a critical exponent.
CFT description The fluctuations at temperature Tc 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 this context, G(r) is understood as a correlation function of scalar fields: It has been suggested that this article or section be merged into Scalar field theory. ...
 Now we can fit together a number of the ideas we've seen already. 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  is modified to become  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
 . 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: In theoretical physics, the term minimal model usually refers to a special class of conformal field theories that generalize the Ising model. ...
 .
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). 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. Relating other 2d CFTs to SLE is an active area of research. 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 one zooms down to finer and finer scales. ...
In probability theory, stochastic Loewner evolution (SLE) is a one-parameter family of random conformally invariant curves in the plane. ...
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ...
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: 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 system. ...
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 Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ...
For other uses, see Liquid (disambiguation). ...
Vapor (US English) or vapour (British English) is the gaseous state of matter. ...
The term critical point can mean any of: critical point (thermodynamics) critical point (mathematics) critical loops (topology) critical point (set theory) This is a disambiguation page: a list of articles associated with the same title. ...
The set of different microscopic theories described by the same scale-invariant theory is known as a universality class. Other examples of systems which belong to a universality class are: In physics, the term renormalization refers to a variety of theoretical concepts and computational techniques revolving either around the idea of rescaling transformations, or around the process of removing infinities from the calculated quantities (see also regularization). ...
- 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, and the phenomenon of diffusion-limited aggregation.
- 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. The toe of an avalanche in Alaskas Kenai Fjords. ...
Definition An interruption in availability of the system due to the communication failure of the network. ...
The term electrical breakdown has several similar but distinctly different meanings. ...
A dielectric is a nonconducting substance, i. ...
In chemistry and other physical sciences, percolation is a type of filtering. ...
Petro redirects here. ...
For the Second Person album, see Chromatography (album). ...
diffusion (disambiguation). ...
3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ...
Making a saline water solution by dissolving table salt (NaCl) in water This article is about chemical solutions. ...
A DLA cluster grown from a copper sulfate solution in an electrodeposition cell 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 diagram shows the nomenclature for the different phase transitions. ...
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. 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. For a Newtonian fluid these take the respective forms This box: Fluid mechanics is the study of how fluids move and the forces on them. ...
In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. ...
All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...
A Newtonian fluid (named for Isaac Newton) is a fluid that flows like water—its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ...
  where μ is the dynamic viscosity. Viscosity is a measure of the resistance of a fluid to deformation under shear 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. 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 In physics and thermodynamics, an equation of state is a relation between state variables. ...
An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. ...
An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces, where the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container and each other and are in constant random motion. ...
 where cs 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.
Computer vision In computer vision, scale invariance refers to a local image description that remains invariant when the scale of the image is changed. 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. Examples of scale invariant blob detectors and ridge detectors are given in the articles on blob detection and ridge detection. An example of the application of scale invariance to object recognition is given in the article on the scale-invariant feature transform. Computer vision is the science and technology of machines that see. ...
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities. ...
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 than the surrounding. ...
In a 2-D function, a (bright) ridge is a connected set of points that are maximal in at least one dimension. ...
Scale-invariant feature transform (or SIFT) is a computer vision algorithm for extracting distinctive features from images, to be used in algorithms for tasks like matching different views of an object or scene (e. ...
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.
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