4190:, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed. Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data. Examples of applications include
1569:
3624:
3485:
1403:
2552:
The φ 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
2628:
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,
3241:
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
2215:
3246:, 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.
1840:
3217:
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:
1185:
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
3696:
1193:
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,
2540:, 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
1641:
1564:{\displaystyle {\begin{aligned}\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}}}\end{aligned}}}
3277:
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
2874:
144:
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
3619:{\displaystyle \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)}
4113:
3979:
3092:
1327:
4166:
1176:
2366:
771:
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
2414:
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
433:
3189:(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
1592:
1408:
993:
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
2723:
1248:
3932:
747:
275:
108:, 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.
520:
2988:
31:
3170:
2557:. 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
2239: = 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
2114:
4021:
3392:
3889:
2084:
1033:
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,
2932:
1130:
3849:
2305:
2272:
1999:
1911:
1878:
920:
4058:
3429:
1947:
2613:
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
3732:, 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
3784:
3466:
1752:
596:
2765:
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
134:, 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
58:
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.
3274:
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.
338:
3631:
1094:
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,
4606:
Extensive discussion of scale invariance in quantum and statistical field theories, applications to critical phenomena and the epsilon expansion and related topics.
3814:
2746:
2470:
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
1587:
3719:
2427:
A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since
2450:(which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles
645:. 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.
2654:
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
3260:
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.
981:
noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a
2797:
3816:
is the speed of sound in the fluid. Given this equation of state, Navier–Stokes and the continuity equation are invariant under the transformations
1746:
Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,
2504:
is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the
Gaussian fixed point. One example is the
4063:
1949:
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
3938:
3035:
1272:
814:
under additive and reproductive convolution as well as under scale transformation. These include a number of common distributions: the
4118:
1136:
2311:
1391:
353:
2100:, there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless
4330:
Kendal, W. S.; Jørgensen, B. (2011). "Taylor's power law and fluctuation scaling explained by a central-limit-like convergence".
3109:
On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the
3722:
3326:
The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
2664:
119:. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
1208:
961:
further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest
3895:
692:
2770:
2651:
the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.
2467:
2443:
223:
4434:
4259:
2210:{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}-\nabla ^{2}\varphi +g\varphi ^{3}=0.}
1339:
We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will
463:
4464:
4203:
2943:
4657:
3214:
3123:
141:
3987:
3358:
4647:
3855:
3243:
131:
17:
579:. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of
4275:
Eisler, Z.; Bartos, I.; Kertész, J. (2008). "Fluctuation scaling in complex systems: Taylor's law and beyond".
2045:
93:, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the
2896:
2625:, and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)
1100:
3822:
3320:
3186:
2278:
2245:
1963:
1884:
1851:
859:
4534:
Lindeberg, T. (2013) Invariance of visual operations at the level of receptive fields, PLoS ONE 8(7):e66990.
4026:
3397:
1919:
802:
62:
4652:
4642:
4584:
2541:
1030:
958:
3473:
1835:{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}-\nabla ^{2}\varphi =0,}
3745:
3469:
3434:
982:
4585:"Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701-713.
3249:
The set of different microscopic theories described by the same scale-invariant theory is known as a
2639:
1950:
623:, and even then a translation and rotation may have to be applied to match the fractal up to itself.
3102:=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that
2431:
are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.
2225:
4379:
4220:
3335:
2582:
2403:
depend on the energy-scale of a given physical process. This energy dependence is described by the
1056:, is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of
807:
773:
149:
135:
94:
4432:
Jørgensen, B.; Martinez, J. R.; Tsao, M. (1994). "Asymptotic behaviour of the variance function".
3691:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {u} \right)=0}
4215:
3110:
2490:
2439:
2408:
1045:
4546:
2605:
An example that links together many of the ideas in this article is the phase transition of the
301:
2521:
1384:
1080:
1068:
826:, as well as more unusual distributions like the compound Poisson-gamma distribution, positive
649:
560:
170:
160:, which are scale-invariant statistics of a variable, while the unstandardized moments are not.
105:
82:
1636:{\displaystyle {\begin{aligned}x\rightarrow \lambda x,\\t\rightarrow \lambda t.\end{aligned}}}
934:
are positive constants. This variance to mean power law is known in the physics literature as
648:
Some fractals may have multiple scaling factors at play at once; such scaling is studied with
2574:
2558:
2416:
2404:
1356:
1002:
972:
950:
946:
811:
796:
559:
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a
123:
3355:
is a scale-invariant classical field theory. The fields are the velocity of the fluid flow,
1013:, the Tweedie convergence theorem requires certain non-Gaussian random variables to express
998:, and the Tweedie distributions become foci of convergence for a wide range of data types.
4500:
4398:
4339:
4296:
3792:
3330:
The key observation is that, for all of these different systems, the behaviour resembles a
3316:
3282:
2788:
2731:
2471:
2396:
2375:
must be dimensionless, otherwise one introduces a fixed length scale into the theory: For
1344:
995:
819:
784:
580:
572:
564:
289:
112:
3704:
977:
provides a hypothetical explanation for the wide manifestation of fluctuation scaling and
830:, and extreme stable distributions. Consequent to their inherent scale invariance Tweedie
8:
4621:
Statistical Field Theory. An
Introduction to Exactly Solved Models of Statistical Physics
4277:
2562:
2027:, Δ, has not been so important. However, one usually requires that the scalar field
1709:
1006:
827:
815:
780:
576:
157:
4504:
4402:
4343:
4300:
2589:
spatial dimensions, the corresponding statistical field theory is formally similar to a
27:
Features that do not change if length or energy scales are multiplied by a common factor
4566:
4516:
4490:
4443:
4312:
4286:
4230:
3250:
3181:
The anomalous dimensions in certain two-dimensional CFTs can be related to the typical
2525:
2517:
1026:
823:
439:
74:
2434:
However, in nature the electromagnetic field is coupled to charged particles, such as
4637:
4520:
4414:
4355:
4316:
4255:
4187:
3729:
3182:
2594:
2593:-dimensional CFT. The scaling dimensions in such problems are usually referred to as
2554:
2533:
2529:
2400:
2101:
2032:
2024:
1333:
1182:
679:
443:
4558:
4508:
4460:
4406:
4347:
4304:
4195:
3331:
2869:{\displaystyle \langle \phi (0)\phi (r)\rangle \propto {\frac {1}{r^{D-2+\eta }}}.}
2578:
2383: = 4. Note that under these transformations the argument of the function
2028:
1713:
1057:
568:
127:
116:
4570:
4533:
4199:
4183:
3477:
3352:
2618:
2447:
1005:
requires certain kinds of random variables to have as a focus of convergence the
954:
939:
831:
765:
90:
4512:
4410:
4351:
4191:
3304:. Power-law scaling connects the rate of flow to the distribution of fractures.
3301:
3264:
2622:
1088:
761:
98:
81:
In mathematics, scale invariance usually refers to an invariance of individual
35:
4562:
4308:
1359:
with no charges or currents. The fields are the electric and magnetic fields,
4631:
1395:
945:
Random sequences, governed by the
Tweedie distributions and evaluated by the
806:, a class of statistical models used to describe error distributions for the
584:
1708:
Another example of a scale-invariant classical field theory is the massless
206:, which can be taken to be a length or size rescaling. The requirement for
4418:
4359:
2597:, and one can in principle compute these exponents in the appropriate CFT.
656:
619:. A fractal is equal to itself typically for only a discrete set of values
616:
615:
are scale-invariant, although more precisely, one should say that they are
604:
2438:. The QFT describing the interactions of photons and charged particles is
525:
Allowing for rotations of the curve, it is invariant under all rescalings
138:, and are formally very similar to scale-invariant quantum field theories.
30:
4177:
3728:
In order to deduce the scale invariance of these equations we specify an
3293:
3223:
3198:
2610:
2606:
1049:
1010:
753:
47:
4108:{\displaystyle \lambda \mathbf {u} (\lambda \mathbf {x} ,\lambda ^{2}t)}
3098:
In the physically interesting case of three spatial dimensions, we have
2442:(QED), and this theory is not scale-invariant. We can see this from the
4478:
4447:
3733:
3300:
through fractured rock beds, or water through filter paper, such as in
3286:
3008:
as a particular critical exponent of the Ising model phase transition.
2004:
and so it should not be surprising that massive scalar field theory is
962:
757:
627:
600:
153:
51:
3253:. Other examples of systems which belong to a universality class are:
2617:-dimensional periodic lattice. Associated with each lattice site is a
4380:"Tweedie convergence: A mathematical basis for Taylor's power law, 1/
4225:
3736:
3334:, and that the language of statistical mechanics and scale-invariant
3308:
3297:
3257:
986:
3346:
3312:
3268:
2435:
990:
989:
of a
Tweedie model. Almost all distribution functions with finite
838:
296:
4495:
4291:
3974:{\displaystyle \mathbf {u} \rightarrow \lambda ^{-1}\mathbf {u} .}
3113:, a family of well-understood CFTs, and it is possible to compute
3087:{\displaystyle \eta ={\frac {\epsilon ^{2}}{54}}+O(\epsilon ^{3})}
1322:{\displaystyle \varphi (x)=\lambda ^{-\Delta }\varphi (\lambda x)}
953:
relationship between the variance to mean power law and power law
4161:{\displaystyle \lambda \rho (\lambda \mathbf {x} ,\lambda ^{2}t)}
1343:
be scale-invariant, and in such cases the symmetry is said to be
1171:{\displaystyle \varphi \rightarrow \lambda ^{-\Delta }\varphi ~.}
612:
43:
2516:
Scale-invariant QFTs are almost always invariant under the full
2361:{\displaystyle \varphi (x)\rightarrow \lambda ^{-1}\varphi (x).}
3234:
3230:
2890:. This is because the classical dimension of the scalar field,
2428:
2020:
428:{\displaystyle f(\lambda x)=(\lambda x)^{n}=\lambda ^{n}f(x)~.}
1581:
These field equations are invariant under the transformation
1063:
174:
169:
In mathematics, one can consider the scaling properties of a
164:
86:
2879:
Now we can fit together a number of the ideas seen already.
1079:. Valid field configurations are then determined by solving
217:
to be invariant under all rescalings is usually taken to be
101:
to display this kind of scale invariance or self-similarity.
2390:
1253:
846:
662:
4609:
1355:
An example of a scale-invariant classical field theory is
595:
280:
for some choice of exponent Δ, and for all dilations
2585:. For a system in equilibrium (i.e. time-independent) in
1957:
is physically equivalent to a fixed length scale through
130:. The key observation is that near a phase transition or
3185:
of random walks, where the random walks are defined via
2997:
is the number of dimensions of the Ising model lattice.
1071:
is generically described by a field, or set of fields,
1029:, the power spectrum of the spatial distribution of the
790:
2718:{\displaystyle G(r)\propto {\frac {1}{r^{D-2+\eta }}},}
2093:
is the combined number of spatial and time dimensions.
2882:
From the above, one sees that the critical exponent,
2581:, its fluctuations are described by a scale-invariant
1243:{\displaystyle \lambda ^{\Delta }\varphi (\lambda x).}
4431:
4121:
4066:
4029:
3990:
3941:
3898:
3858:
3825:
3795:
3748:
3707:
3634:
3488:
3437:
3400:
3361:
3341:
3126:
3038:
2946:
2899:
2800:
2734:
2667:
2314:
2281:
2248:
2117:
2048:
1966:
1922:
1887:
1854:
1755:
1716:
is unrelated to scale invariance). The scalar field,
1590:
1406:
1275:
1211:
1139:
1103:
862:
695:
466:
356:
304:
226:
115:, scale invariance has an interpretation in terms of
3927:{\displaystyle \rho \rightarrow \lambda ^{-1}\rho ,}
742:{\displaystyle P(f)=\lambda ^{-\Delta }P(\lambda f)}
567:. Homogeneous functions are the natural denizens of
4610:DiFrancesco, P.; Mathieu, P.; Senechal, D. (1997).
4274:
442:, a kind of curve that often appears in nature. In
270:{\displaystyle f(\lambda x)=\lambda ^{\Delta }f(x)}
4547:"Feature detection with automatic scale selection"
4160:
4107:
4052:
4015:
3973:
3926:
3883:
3843:
3808:
3778:
3713:
3690:
3618:
3460:
3423:
3386:
3164:
3086:
2982:
2926:
2868:
2740:
2717:
2360:
2299:
2266:
2209:
2078:
2019:The field equations in the examples above are all
1993:
1941:
1916:The name massless refers to the absence of a term
1905:
1872:
1834:
1646:Moreover, given solutions of Maxwell's equations,
1635:
1563:
1321:
1242:
1170:
1124:
914:
779:Examples of scale-invariant distributions are the
741:
514:
427:
332:
269:
2461:
2419:of the corresponding renormalization group flow.
4629:
4465:"Critical Phenomena: field theoretical approach"
3347:Newtonian fluid mechanics with no applied forces
2547:
515:{\displaystyle \theta ={\frac {1}{b}}\ln(r/a)~.}
4544:
4476:
4377:
4329:
2983:{\displaystyle \Delta ={\frac {D-2+\eta }{2}},}
2561:, as long as the theory can be studied through
1703:
2748:, which is an example of a critical exponent.
1202:), one always has other solutions of the form
295:Examples of scale-invariant functions are the
191:. That is, one is interested in the shape of
152:are scale-invariant. The analogous concept in
3165:{\displaystyle \eta _{_{D=2}}={\frac {1}{4}}}
1733:is a function of a set of spatial variables,
1350:
571:, and homogeneous polynomials are studied as
438:An example of a scale-invariant curve is the
4577:
4538:
4425:
4373:
4371:
4369:
4243:
4016:{\displaystyle \mathbf {u} (\mathbf {x} ,t)}
3387:{\displaystyle \mathbf {u} (\mathbf {x} ,t)}
3296:of fluids through disordered media, such as
3176:
3117:(and the other critical exponents) exactly,
2828:
2801:
2477:However, even though the classical massless
4602:Quantum Field Theory and Critical Phenomena
4599:
3884:{\displaystyle t\rightarrow \lambda ^{2}t,}
3205:CFTs to SLE is an active area of research.
3193:critical Ising model and the more general 2
3025:can be calculated approximately, using the
1266:), to be scale-invariant, we require that
634:, but the scaling holds only for values of
4323:
3106:is numerically small in three dimensions.
2489:scale-invariant. We can see this from the
2422:
1845:and is invariant under the transformation
1064:Scale invariance in classical field theory
583:, it has connections to various topics in
165:Scale-invariant curves and self-similarity
4494:
4366:
4290:
4268:
4249:
2511:
2485: = 4, the quantized version is
2079:{\displaystyle \Delta ={\frac {D-2}{2}},}
892:
868:
73:). Dilatations can form part of a larger
4618:
4551:International Journal of Computer Vision
4527:
2927:{\displaystyle \Delta ={\frac {D-2}{2}}}
2886:, for this phase transition, is also an
2642:is said to occur. This means that below
2391:Scale invariance in quantum field theory
2023:in the fields, which has meant that the
1254:Scale invariance of field configurations
1125:{\displaystyle x\rightarrow \lambda x~,}
663:Scale invariance in stochastic processes
594:
29:
3844:{\displaystyle x\rightarrow \lambda x,}
2300:{\displaystyle t\rightarrow \lambda t,}
2267:{\displaystyle x\rightarrow \lambda x,}
1994:{\displaystyle L={\frac {\hbar }{mc}},}
1906:{\displaystyle t\rightarrow \lambda t.}
1873:{\displaystyle x\rightarrow \lambda x,}
1258:For a particular field configuration,
915:{\displaystyle {\text{var}}\,(Y)=a^{p}}
14:
4630:
2399:(QFT) is characterised by the way its
2108: = 4. The field equation is
554:
4378:Kendal, W. S.; Jørgensen, B. (2011).
4053:{\displaystyle \rho (\mathbf {x} ,t)}
3468:. These fields must satisfy both the
3424:{\displaystyle \rho (\mathbf {x} ,t)}
3271:, as a function of size and duration.
3004:in the conformal field theory is the
2228:, which contains the fourth power of
2031:is dimensionless, and this fixes the
1942:{\displaystyle \propto m^{2}\varphi }
791:Scale-invariant Tweedie distributions
540:is identical to a rotated version of
2568:
2371:The key point is that the parameter
2500:Even though the quantized massless
1383:), while their field equations are
1087:, and these equations are known as
938:, and in the ecology literature as
126:, scale invariance is a feature of
24:
4593:
4171:
3658:
3646:
3638:
3595:
3587:
3560:
3542:
3528:
3505:
3495:
3342:Other examples of scale invariance
3289:, which resemble cracks and tears.
3226:phase transition, described above.
2947:
2900:
2751:
2658:. This has the generic behaviour:
2600:
2173:
2153:
2139:
2049:
1811:
1791:
1777:
1541:
1525:
1487:
1466:
1450:
1412:
1299:
1217:
1154:
719:
250:
25:
4669:
4204:scale-invariant feature transform
4202:, and object recognition via the
3779:{\displaystyle P=c_{s}^{2}\rho ,}
3461:{\displaystyle P(\mathbf {x} ,t)}
3338:may be applied to describe them.
2379:theory, this is only the case in
2096:Given this scaling dimension for
187:under rescalings of the variable
4135:
4082:
4071:
4037:
4000:
3992:
3964:
3943:
3673:
3602:
3570:
3532:
3521:
3499:
3480:these take the respective forms
3445:
3408:
3371:
3363:
2756:The fluctuations at temperature
2520:, and the study of such QFTs is
1535:
1497:
1460:
1422:
1181:The parameter Δ is known as the
4252:The Theory of Dispersion Models
3237:transition in classical fluids.
3208:
2769:, a particular scale-invariant
1017:noise and fluctuation scaling.
457:, the spiral can be written as
89:. A closely related concept is
4477:Salvio, Strumia (2014-03-17).
4470:
4454:
4254:. London: Chapman & Hall.
4155:
4128:
4102:
4075:
4047:
4033:
4010:
3996:
3947:
3902:
3862:
3829:
3455:
3441:
3418:
3404:
3381:
3367:
3081:
3068:
2825:
2819:
2813:
2807:
2677:
2671:
2352:
2346:
2327:
2324:
2318:
2285:
2252:
1891:
1858:
1617:
1598:
1316:
1307:
1285:
1279:
1234:
1225:
1143:
1107:
1075:, that depend on coordinates,
903:
899:
893:
884:
875:
869:
736:
727:
705:
699:
503:
489:
416:
410:
385:
375:
369:
360:
314:
308:
264:
258:
239:
230:
13:
1:
4463:(2010) Scholarpedia article
4236:
4060:, we automatically have that
3351:Under certain circumstances,
3321:diffusion-limited aggregation
2728:for some particular value of
2548:Scale and conformal anomalies
2528:in a CFT have a well-defined
2481:theory is scale-invariant in
2468:quantized scalar field theory
2224:derives from the form of the
1390:With no charges or currents,
991:cumulant generating functions
803:exponential dispersion models
2542:anomalous scaling dimensions
2493:for the coupling parameter,
2462:Massless scalar field theory
2011:
1704:Massless scalar field theory
1020:
760:, and Δ = −2 for
61:The technical term for this
7:
4384:noise, and multifractality"
4209:
1951:relativistic field theories
1031:cosmic microwave background
974:Tweedie convergence theorem
590:
10:
4674:
4623:. Oxford University Press.
4604:. Oxford University Press.
4600:Zinn-Justin, Jean (2002).
4411:10.1103/PhysRevE.84.066120
4352:10.1103/PhysRevE.83.066115
4175:
3431:, and the fluid pressure,
2577:, as a system undergoes a
2395:The scale-dependence of a
1351:Classical electromagnetism
752:with Δ = 0 for
657:external and internal rays
611:It is sometimes said that
563:, and more generally to a
333:{\displaystyle f(x)=x^{n}}
136:statistical field theories
4309:10.1080/00018730801893043
3187:Schramm–Loewner evolution
3177:Schramm–Loewner evolution
2767:Wilson–Fisher fixed point
2640:spontaneous magnetization
2506:Wilson–Fisher fixed point
2446:. This tells us that the
756:, Δ = −1 for
95:probability distributions
4545:Lindeberg, Tony (1998).
4221:Inverse square potential
3336:statistical field theory
3319:, and the phenomenon of
3011:Note that for dimension
2583:statistical field theory
2454:scale-invariant, QED is
2407:, and is encoded in the
947:method of expanding bins
808:generalized linear model
774:probability distribution
284:. This is equivalent to
150:dimensionless quantities
4658:Scale-invariant systems
4563:10.1023/A:1008045108935
4513:10.1007/JHEP06(2014)080
4216:Invariant (mathematics)
2440:quantum electrodynamics
2423:Quantum electrodynamics
1739:, and a time variable,
1700:) are also solutions.
1578:is the speed of light.
1332:where Δ is, again, the
1046:primordial fluctuations
959:Wiener–Khinchin theorem
686:, then noise scales as
659:are invariant curves .
626:Thus, for example, the
4648:Conformal field theory
4612:Conformal Field Theory
4250:Jørgensen, B. (1997).
4162:
4109:
4054:
4017:
3975:
3928:
3885:
3845:
3810:
3780:
3715:
3692:
3620:
3470:Navier–Stokes equation
3462:
3425:
3388:
3213:A phenomenon known as
3166:
3088:
2984:
2937:is modified to become
2928:
2870:
2742:
2719:
2522:conformal field theory
2512:Conformal field theory
2362:
2301:
2268:
2211:
2080:
1995:
1943:
1907:
1874:
1836:
1637:
1565:
1323:
1244:
1172:
1126:
1081:differential equations
1069:Classical field theory
985:that comes within the
916:
800:are a special case of
743:
650:multi-fractal analysis
608:
561:homogeneous polynomial
516:
429:
334:
271:
202:for some scale factor
106:classical field theory
39:
4619:Mussardo, G. (2010).
4163:
4110:
4055:
4018:
3976:
3929:
3886:
3846:
3811:
3809:{\displaystyle c_{s}}
3781:
3716:
3693:
3621:
3463:
3426:
3394:, the fluid density,
3389:
3167:
3089:
3029:, and one finds that
2985:
2929:
2871:
2743:
2741:{\displaystyle \eta }
2720:
2575:statistical mechanics
2405:renormalization group
2363:
2302:
2269:
2212:
2081:
1996:
1944:
1908:
1875:
1837:
1638:
1566:
1392:these field equations
1324:
1245:
1173:
1127:
1007:Gaussian distribution
1003:central limit theorem
917:
810:and characterized by
797:Tweedie distributions
764:(and more generally,
744:
598:
517:
430:
335:
272:
124:statistical mechanics
33:
4583:T. Lindeberg (2014)
4168:are also solutions.
4119:
4064:
4027:
3988:
3984:Given the solutions
3939:
3896:
3856:
3823:
3793:
3746:
3714:{\displaystyle \mu }
3705:
3632:
3486:
3435:
3398:
3359:
3283:electrical breakdown
3124:
3036:
2944:
2897:
2798:
2789:correlation function
2732:
2665:
2609:, a simple model of
2472:Gaussian fixed point
2397:quantum field theory
2312:
2279:
2246:
2220:(Note that the name
2115:
2046:
1964:
1920:
1885:
1852:
1753:
1712:(note that the name
1588:
1404:
1345:spontaneously broken
1273:
1209:
1137:
1101:
987:domain of attraction
860:
828:stable distributions
820:Poisson distribution
785:Zipfian distribution
693:
573:projective varieties
565:homogeneous function
464:
354:
302:
290:homogeneous function
224:
158:standardized moments
113:quantum field theory
4505:2014JHEP...06..080S
4403:2011PhRvE..84f6120K
4344:2011PhRvE..83f6115K
4301:2008AdPhy..57...89E
3769:
3474:continuity equation
3002:anomalous dimension
2888:anomalous dimension
2787:is understood as a
2771:scalar field theory
2563:perturbation theory
2532:, analogous to the
2401:coupling parameters
1670:), it holds that
1385:Maxwell's equations
996:asymptotic behavior
936:fluctuation scaling
816:normal distribution
781:Pareto distribution
682:power at frequency
577:projective geometry
555:Projective geometry
4653:Critical phenomena
4643:Scaling symmetries
4614:. Springer-Verlag.
4231:Scale-free network
4158:
4105:
4050:
4013:
3971:
3924:
3881:
3841:
3806:
3776:
3755:
3739:, which satisfies
3711:
3688:
3616:
3458:
3421:
3384:
3251:universality class
3201:. Relating other 2
3183:fractal dimensions
3162:
3084:
2980:
2924:
2866:
2791:of scalar fields,
2738:
2715:
2595:critical exponents
2518:conformal symmetry
2358:
2297:
2264:
2207:
2076:
1991:
1939:
1903:
1870:
1832:
1633:
1631:
1561:
1559:
1319:
1240:
1168:
1122:
1027:physical cosmology
912:
824:gamma distribution
739:
609:
512:
440:logarithmic spiral
425:
347:, in that clearly
330:
292:of degree Δ.
267:
75:conformal symmetry
40:
38:is scale-invariant
4188:biological vision
3730:equation of state
3723:dynamic viscosity
3653:
3585:
3512:
3263:The frequency of
3160:
3060:
3027:epsilon expansion
2975:
2922:
2861:
2776:In this context,
2710:
2569:Phase transitions
2534:scaling dimension
2530:scaling dimension
2458:scale-invariant.
2444:QED beta-function
2167:
2133:
2071:
2039:. In particular,
2033:scaling dimension
2025:scaling dimension
2008:scale-invariant.
1986:
1805:
1771:
1555:
1519:
1480:
1444:
1394:take the form of
1334:scaling dimension
1190:scale-invariant.
1183:scaling dimension
1164:
1118:
1048:as a function of
983:variance function
890:
866:
680:average, expected
508:
481:
444:polar coordinates
421:
128:phase transitions
16:(Redirected from
4665:
4624:
4615:
4605:
4587:
4581:
4575:
4574:
4542:
4536:
4531:
4525:
4524:
4498:
4474:
4468:
4458:
4452:
4451:
4429:
4423:
4422:
4388:
4375:
4364:
4363:
4327:
4321:
4320:
4294:
4272:
4266:
4265:
4247:
4196:corner detection
4167:
4165:
4164:
4159:
4151:
4150:
4138:
4114:
4112:
4111:
4106:
4098:
4097:
4085:
4074:
4059:
4057:
4056:
4051:
4040:
4022:
4020:
4019:
4014:
4003:
3995:
3980:
3978:
3977:
3972:
3967:
3962:
3961:
3946:
3933:
3931:
3930:
3925:
3917:
3916:
3890:
3888:
3887:
3882:
3874:
3873:
3850:
3848:
3847:
3842:
3815:
3813:
3812:
3807:
3805:
3804:
3785:
3783:
3782:
3777:
3768:
3763:
3720:
3718:
3717:
3712:
3697:
3695:
3694:
3689:
3681:
3677:
3676:
3654:
3652:
3644:
3636:
3625:
3623:
3622:
3617:
3615:
3611:
3610:
3606:
3605:
3586:
3578:
3573:
3568:
3567:
3535:
3524:
3513:
3511:
3503:
3502:
3493:
3467:
3465:
3464:
3459:
3448:
3430:
3428:
3427:
3422:
3411:
3393:
3391:
3390:
3385:
3374:
3366:
3332:phase transition
3171:
3169:
3168:
3163:
3161:
3153:
3148:
3147:
3146:
3145:
3116:
3105:
3101:
3093:
3091:
3090:
3085:
3080:
3079:
3061:
3056:
3055:
3046:
3024:
3020:
2996:
2989:
2987:
2986:
2981:
2976:
2971:
2954:
2933:
2931:
2930:
2925:
2923:
2918:
2907:
2885:
2875:
2873:
2872:
2867:
2862:
2860:
2859:
2835:
2786:
2764:
2747:
2745:
2744:
2739:
2724:
2722:
2721:
2716:
2711:
2709:
2708:
2684:
2657:
2650:
2637:
2616:
2592:
2588:
2579:phase transition
2559:cosmic inflation
2386:
2382:
2378:
2374:
2367:
2365:
2364:
2359:
2342:
2341:
2306:
2304:
2303:
2298:
2273:
2271:
2270:
2265:
2238:
2231:
2223:
2216:
2214:
2213:
2208:
2200:
2199:
2181:
2180:
2168:
2166:
2165:
2164:
2151:
2147:
2146:
2136:
2134:
2132:
2131:
2119:
2107:
2099:
2092:
2085:
2083:
2082:
2077:
2072:
2067:
2056:
2038:
2000:
1998:
1997:
1992:
1987:
1985:
1974:
1956:
1953:, a mass-scale,
1948:
1946:
1945:
1940:
1935:
1934:
1912:
1910:
1909:
1904:
1879:
1877:
1876:
1871:
1841:
1839:
1838:
1833:
1819:
1818:
1806:
1804:
1803:
1802:
1789:
1785:
1784:
1774:
1772:
1770:
1769:
1757:
1742:
1732:
1642:
1640:
1639:
1634:
1632:
1570:
1568:
1567:
1562:
1560:
1556:
1554:
1553:
1552:
1539:
1538:
1533:
1532:
1522:
1520:
1518:
1517:
1505:
1500:
1495:
1494:
1481:
1479:
1478:
1477:
1464:
1463:
1458:
1457:
1447:
1445:
1443:
1442:
1430:
1425:
1420:
1419:
1357:electromagnetism
1328:
1326:
1325:
1320:
1303:
1302:
1249:
1247:
1246:
1241:
1221:
1220:
1177:
1175:
1174:
1169:
1162:
1158:
1157:
1131:
1129:
1128:
1123:
1116:
1058:cosmic inflation
1055:
1043:
955:autocorrelations
921:
919:
918:
913:
911:
910:
891:
888:
867:
864:
832:random variables
748:
746:
745:
740:
723:
722:
685:
677:
644:
640:
633:
622:
569:projective space
550:
539:
528:
521:
519:
518:
513:
506:
499:
482:
474:
456:
434:
432:
431:
426:
419:
406:
405:
393:
392:
346:
339:
337:
336:
331:
329:
328:
288: being a
287:
283:
276:
274:
273:
268:
254:
253:
216:
205:
201:
190:
186:
117:particle physics
99:random processes
56:scale invariance
21:
4673:
4672:
4668:
4667:
4666:
4664:
4663:
4662:
4628:
4627:
4596:
4594:Further reading
4591:
4590:
4582:
4578:
4543:
4539:
4532:
4528:
4475:
4471:
4459:
4455:
4435:Scand J Statist
4430:
4426:
4386:
4376:
4367:
4328:
4324:
4273:
4269:
4262:
4248:
4244:
4239:
4212:
4200:ridge detection
4184:computer vision
4180:
4174:
4172:Computer vision
4146:
4142:
4134:
4120:
4117:
4116:
4093:
4089:
4081:
4070:
4065:
4062:
4061:
4036:
4028:
4025:
4024:
3999:
3991:
3989:
3986:
3985:
3963:
3954:
3950:
3942:
3940:
3937:
3936:
3909:
3905:
3897:
3894:
3893:
3869:
3865:
3857:
3854:
3853:
3824:
3821:
3820:
3800:
3796:
3794:
3791:
3790:
3764:
3759:
3747:
3744:
3743:
3706:
3703:
3702:
3672:
3668:
3664:
3645:
3637:
3635:
3633:
3630:
3629:
3601:
3594:
3590:
3577:
3569:
3563:
3559:
3558:
3554:
3531:
3520:
3504:
3498:
3494:
3492:
3487:
3484:
3483:
3478:Newtonian fluid
3444:
3436:
3433:
3432:
3407:
3399:
3396:
3395:
3370:
3362:
3360:
3357:
3356:
3353:fluid mechanics
3349:
3344:
3265:network outages
3211:
3179:
3152:
3135:
3132:
3131:
3127:
3125:
3122:
3121:
3114:
3103:
3099:
3075:
3071:
3051:
3047:
3045:
3037:
3034:
3033:
3022:
3012:
2994:
2955:
2953:
2945:
2942:
2941:
2908:
2906:
2898:
2895:
2894:
2883:
2843:
2839:
2834:
2799:
2796:
2795:
2777:
2762:
2757:
2754:
2752:CFT description
2733:
2730:
2729:
2692:
2688:
2683:
2666:
2663:
2662:
2655:
2648:
2643:
2635:
2630:
2619:magnetic moment
2614:
2603:
2601:The Ising model
2590:
2586:
2571:
2550:
2514:
2466:Free, massless
2464:
2448:electric charge
2425:
2411:of the theory.
2393:
2384:
2380:
2376:
2372:
2334:
2330:
2313:
2310:
2309:
2280:
2277:
2276:
2247:
2244:
2243:
2236:
2229:
2221:
2195:
2191:
2176:
2172:
2160:
2156:
2152:
2142:
2138:
2137:
2135:
2127:
2123:
2118:
2116:
2113:
2112:
2105:
2097:
2090:
2057:
2055:
2047:
2044:
2043:
2036:
2017:
1978:
1973:
1965:
1962:
1961:
1954:
1930:
1926:
1921:
1918:
1917:
1886:
1883:
1882:
1853:
1850:
1849:
1814:
1810:
1798:
1794:
1790:
1780:
1776:
1775:
1773:
1765:
1761:
1756:
1754:
1751:
1750:
1740:
1717:
1706:
1630:
1629:
1611:
1610:
1591:
1589:
1586:
1585:
1558:
1557:
1548:
1544:
1540:
1534:
1528:
1524:
1523:
1521:
1513:
1509:
1504:
1496:
1490:
1486:
1483:
1482:
1473:
1469:
1465:
1459:
1453:
1449:
1448:
1446:
1438:
1434:
1429:
1421:
1415:
1411:
1407:
1405:
1402:
1401:
1353:
1295:
1291:
1274:
1271:
1270:
1256:
1216:
1212:
1210:
1207:
1206:
1150:
1146:
1138:
1135:
1134:
1102:
1099:
1098:
1089:field equations
1066:
1053:
1034:
1023:
906:
902:
887:
863:
861:
858:
857:
793:
766:Brownian motion
715:
711:
694:
691:
690:
683:
668:
665:
642:
635:
631:
620:
593:
557:
541:
530:
526:
495:
473:
465:
462:
461:
446:
401:
397:
388:
384:
355:
352:
351:
341:
324:
320:
303:
300:
299:
285:
281:
249:
245:
225:
222:
221:
207:
203:
192:
188:
177:
167:
91:self-similarity
69:(also known as
28:
23:
22:
18:Scale invariant
15:
12:
11:
5:
4671:
4661:
4660:
4655:
4650:
4645:
4640:
4626:
4625:
4616:
4607:
4595:
4592:
4589:
4588:
4576:
4537:
4526:
4469:
4461:J. Zinn-Justin
4453:
4442:(3): 223–243.
4424:
4365:
4322:
4267:
4261:978-0412997112
4260:
4241:
4240:
4238:
4235:
4234:
4233:
4228:
4223:
4218:
4211:
4208:
4192:blob detection
4176:Main article:
4173:
4170:
4157:
4154:
4149:
4145:
4141:
4137:
4133:
4130:
4127:
4124:
4104:
4101:
4096:
4092:
4088:
4084:
4080:
4077:
4073:
4069:
4049:
4046:
4043:
4039:
4035:
4032:
4012:
4009:
4006:
4002:
3998:
3994:
3982:
3981:
3970:
3966:
3960:
3957:
3953:
3949:
3945:
3934:
3923:
3920:
3915:
3912:
3908:
3904:
3901:
3891:
3880:
3877:
3872:
3868:
3864:
3861:
3851:
3840:
3837:
3834:
3831:
3828:
3803:
3799:
3787:
3786:
3775:
3772:
3767:
3762:
3758:
3754:
3751:
3710:
3699:
3698:
3687:
3684:
3680:
3675:
3671:
3667:
3663:
3660:
3657:
3651:
3648:
3643:
3640:
3614:
3609:
3604:
3600:
3597:
3593:
3589:
3584:
3581:
3576:
3572:
3566:
3562:
3557:
3553:
3550:
3547:
3544:
3541:
3538:
3534:
3530:
3527:
3523:
3519:
3516:
3510:
3507:
3501:
3497:
3491:
3457:
3454:
3451:
3447:
3443:
3440:
3420:
3417:
3414:
3410:
3406:
3403:
3383:
3380:
3377:
3373:
3369:
3365:
3348:
3345:
3343:
3340:
3328:
3327:
3324:
3305:
3302:chromatography
3290:
3279:
3275:
3272:
3261:
3244:critical point
3239:
3238:
3227:
3210:
3207:
3178:
3175:
3174:
3173:
3159:
3156:
3151:
3144:
3141:
3138:
3134:
3130:
3111:minimal models
3096:
3095:
3083:
3078:
3074:
3070:
3067:
3064:
3059:
3054:
3050:
3044:
3041:
2991:
2990:
2979:
2974:
2970:
2967:
2964:
2961:
2958:
2952:
2949:
2935:
2934:
2921:
2917:
2914:
2911:
2905:
2902:
2877:
2876:
2865:
2858:
2855:
2852:
2849:
2846:
2842:
2838:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2760:
2753:
2750:
2737:
2726:
2725:
2714:
2707:
2704:
2701:
2698:
2695:
2691:
2687:
2682:
2679:
2676:
2673:
2670:
2646:
2633:
2602:
2599:
2570:
2567:
2549:
2546:
2513:
2510:
2463:
2460:
2424:
2421:
2409:beta-functions
2392:
2389:
2387:is unchanged.
2369:
2368:
2357:
2354:
2351:
2348:
2345:
2340:
2337:
2333:
2329:
2326:
2323:
2320:
2317:
2307:
2296:
2293:
2290:
2287:
2284:
2274:
2263:
2260:
2257:
2254:
2251:
2218:
2217:
2206:
2203:
2198:
2194:
2190:
2187:
2184:
2179:
2175:
2171:
2163:
2159:
2155:
2150:
2145:
2141:
2130:
2126:
2122:
2087:
2086:
2075:
2070:
2066:
2063:
2060:
2054:
2051:
2016:
2010:
2002:
2001:
1990:
1984:
1981:
1977:
1972:
1969:
1938:
1933:
1929:
1925:
1914:
1913:
1902:
1899:
1896:
1893:
1890:
1880:
1869:
1866:
1863:
1860:
1857:
1843:
1842:
1831:
1828:
1825:
1822:
1817:
1813:
1809:
1801:
1797:
1793:
1788:
1783:
1779:
1768:
1764:
1760:
1705:
1702:
1644:
1643:
1628:
1625:
1622:
1619:
1616:
1613:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1593:
1572:
1571:
1551:
1547:
1543:
1537:
1531:
1527:
1516:
1512:
1508:
1503:
1499:
1493:
1489:
1485:
1484:
1476:
1472:
1468:
1462:
1456:
1452:
1441:
1437:
1433:
1428:
1424:
1418:
1414:
1410:
1409:
1396:wave equations
1352:
1349:
1336:of the field.
1330:
1329:
1318:
1315:
1312:
1309:
1306:
1301:
1298:
1294:
1290:
1287:
1284:
1281:
1278:
1255:
1252:
1251:
1250:
1239:
1236:
1233:
1230:
1227:
1224:
1219:
1215:
1179:
1178:
1167:
1161:
1156:
1153:
1149:
1145:
1142:
1132:
1121:
1115:
1112:
1109:
1106:
1065:
1062:
1022:
1019:
924:
923:
909:
905:
901:
898:
895:
886:
883:
880:
877:
874:
871:
853:) power law:
837:demonstrate a
792:
789:
762:Brownian noise
750:
749:
738:
735:
732:
729:
726:
721:
718:
714:
710:
707:
704:
701:
698:
664:
661:
592:
589:
556:
553:
523:
522:
511:
505:
502:
498:
494:
491:
488:
485:
480:
477:
472:
469:
436:
435:
424:
418:
415:
412:
409:
404:
400:
396:
391:
387:
383:
380:
377:
374:
371:
368:
365:
362:
359:
327:
323:
319:
316:
313:
310:
307:
278:
277:
266:
263:
260:
257:
252:
248:
244:
241:
238:
235:
232:
229:
166:
163:
162:
161:
146:
139:
132:critical point
120:
109:
102:
63:transformation
36:Wiener process
26:
9:
6:
4:
3:
2:
4670:
4659:
4656:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4636:
4635:
4633:
4622:
4617:
4613:
4608:
4603:
4598:
4597:
4586:
4580:
4572:
4568:
4564:
4560:
4557:(2): 79–116.
4556:
4552:
4548:
4541:
4535:
4530:
4522:
4518:
4514:
4510:
4506:
4502:
4497:
4492:
4488:
4484:
4480:
4473:
4466:
4462:
4457:
4449:
4445:
4441:
4437:
4436:
4428:
4420:
4416:
4412:
4408:
4404:
4400:
4397:(6): 066120.
4396:
4392:
4385:
4383:
4374:
4372:
4370:
4361:
4357:
4353:
4349:
4345:
4341:
4338:(6): 066115.
4337:
4333:
4326:
4318:
4314:
4310:
4306:
4302:
4298:
4293:
4288:
4285:(1): 89–142.
4284:
4280:
4279:
4271:
4263:
4257:
4253:
4246:
4242:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4213:
4207:
4205:
4201:
4197:
4193:
4189:
4185:
4179:
4169:
4152:
4147:
4143:
4139:
4131:
4125:
4122:
4099:
4094:
4090:
4086:
4078:
4067:
4044:
4041:
4030:
4007:
4004:
3968:
3958:
3955:
3951:
3935:
3921:
3918:
3913:
3910:
3906:
3899:
3892:
3878:
3875:
3870:
3866:
3859:
3852:
3838:
3835:
3832:
3826:
3819:
3818:
3817:
3801:
3797:
3773:
3770:
3765:
3760:
3756:
3752:
3749:
3742:
3741:
3740:
3738:
3735:
3731:
3726:
3724:
3708:
3685:
3682:
3678:
3669:
3665:
3661:
3655:
3649:
3641:
3628:
3627:
3626:
3612:
3607:
3598:
3591:
3582:
3579:
3574:
3564:
3555:
3551:
3548:
3545:
3539:
3536:
3525:
3517:
3514:
3508:
3489:
3481:
3479:
3475:
3471:
3452:
3449:
3438:
3415:
3412:
3401:
3378:
3375:
3354:
3339:
3337:
3333:
3325:
3322:
3318:
3314:
3310:
3306:
3303:
3299:
3295:
3291:
3288:
3284:
3280:
3276:
3273:
3270:
3266:
3262:
3259:
3256:
3255:
3254:
3252:
3247:
3245:
3236:
3232:
3228:
3225:
3221:
3220:
3219:
3216:
3206:
3204:
3200:
3196:
3192:
3188:
3184:
3157:
3154:
3149:
3142:
3139:
3136:
3133:
3128:
3120:
3119:
3118:
3112:
3107:
3076:
3072:
3065:
3062:
3057:
3052:
3048:
3042:
3039:
3032:
3031:
3030:
3028:
3019:
3015:
3009:
3007:
3003:
2998:
2977:
2972:
2968:
2965:
2962:
2959:
2956:
2950:
2940:
2939:
2938:
2919:
2915:
2912:
2909:
2903:
2893:
2892:
2891:
2889:
2880:
2863:
2856:
2853:
2850:
2847:
2844:
2840:
2836:
2831:
2822:
2816:
2810:
2804:
2794:
2793:
2792:
2790:
2784:
2780:
2774:
2772:
2768:
2763:
2749:
2735:
2712:
2705:
2702:
2699:
2696:
2693:
2689:
2685:
2680:
2674:
2668:
2661:
2660:
2659:
2652:
2649:
2641:
2636:
2626:
2624:
2620:
2612:
2611:ferromagnetic
2608:
2598:
2596:
2584:
2580:
2576:
2566:
2564:
2560:
2556:
2545:
2543:
2539:
2535:
2531:
2527:
2523:
2519:
2509:
2507:
2503:
2498:
2496:
2492:
2491:beta-function
2488:
2484:
2480:
2475:
2473:
2469:
2459:
2457:
2453:
2449:
2445:
2441:
2437:
2432:
2430:
2420:
2418:
2412:
2410:
2406:
2402:
2398:
2388:
2355:
2349:
2343:
2338:
2335:
2331:
2321:
2315:
2308:
2294:
2291:
2288:
2282:
2275:
2261:
2258:
2255:
2249:
2242:
2241:
2240:
2233:
2227:
2204:
2201:
2196:
2192:
2188:
2185:
2182:
2177:
2169:
2161:
2157:
2148:
2143:
2128:
2124:
2120:
2111:
2110:
2109:
2103:
2094:
2073:
2068:
2064:
2061:
2058:
2052:
2042:
2041:
2040:
2034:
2030:
2026:
2022:
2014:
2009:
2007:
1988:
1982:
1979:
1975:
1970:
1967:
1960:
1959:
1958:
1952:
1936:
1931:
1927:
1923:
1900:
1897:
1894:
1888:
1881:
1867:
1864:
1861:
1855:
1848:
1847:
1846:
1829:
1826:
1823:
1820:
1815:
1807:
1799:
1795:
1786:
1781:
1766:
1762:
1758:
1749:
1748:
1747:
1744:
1738:
1737:
1730:
1726:
1725:
1720:
1715:
1711:
1701:
1699:
1695:
1692:
1688:
1684:
1680:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1626:
1623:
1620:
1614:
1607:
1604:
1601:
1595:
1584:
1583:
1582:
1579:
1577:
1549:
1545:
1529:
1514:
1510:
1506:
1501:
1491:
1474:
1470:
1454:
1439:
1435:
1431:
1426:
1416:
1400:
1399:
1398:
1397:
1393:
1388:
1386:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1348:
1346:
1342:
1337:
1335:
1313:
1310:
1304:
1296:
1292:
1288:
1282:
1276:
1269:
1268:
1267:
1265:
1261:
1237:
1231:
1228:
1222:
1213:
1205:
1204:
1203:
1201:
1197:
1191:
1189:
1184:
1165:
1159:
1151:
1147:
1140:
1133:
1119:
1113:
1110:
1104:
1097:
1096:
1095:
1092:
1090:
1086:
1082:
1078:
1074:
1070:
1061:
1059:
1051:
1047:
1041:
1037:
1032:
1028:
1018:
1016:
1012:
1008:
1004:
999:
997:
992:
988:
984:
980:
976:
975:
969:
967:
965:
960:
956:
952:
951:biconditional
948:
943:
941:
937:
933:
929:
907:
896:
881:
878:
872:
856:
855:
854:
852:
848:
844:
840:
836:
833:
829:
825:
821:
817:
813:
809:
805:
804:
799:
798:
788:
786:
782:
777:
775:
769:
767:
763:
759:
755:
733:
730:
724:
716:
712:
708:
702:
696:
689:
688:
687:
681:
675:
671:
660:
658:
653:
651:
646:
638:
629:
624:
618:
614:
606:
602:
597:
588:
586:
585:string theory
582:
578:
574:
570:
566:
562:
552:
548:
544:
537:
533:
509:
500:
496:
492:
486:
483:
478:
475:
470:
467:
460:
459:
458:
454:
450:
445:
441:
422:
413:
407:
402:
398:
394:
389:
381:
378:
372:
366:
363:
357:
350:
349:
348:
345:
325:
321:
317:
311:
305:
298:
293:
291:
261:
255:
246:
242:
236:
233:
227:
220:
219:
218:
214:
210:
199:
195:
184:
180:
176:
172:
159:
155:
151:
147:
143:
140:
137:
133:
129:
125:
121:
118:
114:
110:
107:
103:
100:
96:
92:
88:
84:
80:
79:
78:
76:
72:
68:
64:
59:
57:
53:
49:
45:
37:
32:
19:
4620:
4611:
4601:
4579:
4554:
4550:
4540:
4529:
4486:
4482:
4472:
4456:
4439:
4433:
4427:
4394:
4391:Phys. Rev. E
4390:
4381:
4335:
4332:Phys. Rev. E
4331:
4325:
4282:
4276:
4270:
4251:
4245:
4181:
3983:
3788:
3727:
3700:
3482:
3350:
3329:
3248:
3240:
3215:universality
3212:
3209:Universality
3202:
3194:
3190:
3180:
3108:
3097:
3026:
3017:
3013:
3010:
3005:
3001:
2999:
2992:
2936:
2887:
2881:
2878:
2782:
2778:
2775:
2766:
2758:
2755:
2727:
2653:
2644:
2631:
2627:
2604:
2572:
2551:
2537:
2515:
2505:
2501:
2499:
2494:
2486:
2482:
2478:
2476:
2465:
2455:
2451:
2433:
2426:
2417:fixed points
2413:
2394:
2370:
2234:
2219:
2095:
2088:
2018:
2012:
2005:
2003:
1915:
1844:
1745:
1735:
1734:
1728:
1723:
1722:
1718:
1710:scalar field
1707:
1697:
1693:
1690:
1686:
1682:
1678:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1645:
1580:
1575:
1573:
1389:
1380:
1376:
1372:
1368:
1364:
1360:
1354:
1340:
1338:
1331:
1263:
1259:
1257:
1199:
1195:
1192:
1187:
1180:
1093:
1084:
1076:
1072:
1067:
1039:
1035:
1024:
1014:
1009:and express
1001:Much as the
1000:
978:
973:
970:
963:
944:
940:Taylor's law
935:
931:
927:
925:
850:
842:
834:
801:
795:
794:
778:
770:
751:
673:
669:
666:
654:
647:
641:for integer
636:
630:scales with
625:
617:self-similar
610:
605:self-similar
558:
546:
542:
535:
531:
524:
452:
448:
437:
343:
340:, for which
294:
279:
212:
208:
197:
193:
182:
178:
168:
148:In general,
142:Universality
70:
66:
60:
55:
41:
4178:Scale space
3294:percolation
3287:dielectrics
3224:Ising model
3199:Potts model
2607:Ising model
1050:wave number
1011:white noise
754:white noise
529:; that is,
48:mathematics
4632:Categories
4489:(6): 080.
4479:"Agravity"
4237:References
3734:isothermal
3258:Avalanches
2226:Lagrangian
949:exhibit a
758:pink noise
628:Koch curve
601:Koch curve
154:statistics
67:dilatation
52:statistics
4521:256010671
4496:1403.4226
4317:119608542
4292:0708.2053
4226:Power law
4144:λ
4132:λ
4126:ρ
4123:λ
4091:λ
4079:λ
4068:λ
4031:ρ
3956:−
3952:λ
3948:→
3919:ρ
3911:−
3907:λ
3903:→
3900:ρ
3867:λ
3863:→
3833:λ
3830:→
3771:ρ
3737:ideal gas
3709:μ
3670:ρ
3662:⋅
3659:∇
3647:∂
3642:ρ
3639:∂
3599:⋅
3596:∇
3588:∇
3561:∇
3552:μ
3543:∇
3540:−
3529:∇
3526:⋅
3518:ρ
3506:∂
3496:∂
3490:ρ
3402:ρ
3313:molecules
3309:diffusion
3298:petroleum
3197:critical
3129:η
3073:ϵ
3049:ϵ
3040:η
2969:η
2960:−
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2436:electrons
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4278:Adv Phys
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783:and the
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