View source for Dynamic scaling

'''Dynamic scaling''' (sometimes known as '''Family–Vicsek scaling'''<ref>{{cite journal|first1=F.|last1=Family|author1-link=F. Family|first2=T.|last2=Vicsek|author2-link=T. Vicsek|title=Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model|journal=Journal of Physics A: Mathematical and General|volume=18|issue=2|pages=L75–L81|year=1985|doi=10.1088/0305-4470/18/2/005|bibcode=1985JPhA...18L..75F}}</ref><ref name=Vicsek1984>{{cite journal | last1=Vicsek | first1=Tamás | last2=Family | first2=Fereydoon | title=Dynamic Scaling for Aggregation of Clusters | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=52 | issue=19 | date=1984-05-07 | issn=0031-9007 | doi=10.1103/physrevlett.52.1669 | pages=1669–1672| bibcode=1984PhRvL..52.1669V }}</ref>) is a litmus test that shows whether an evolving system exhibits [[self-similarity]]. In general a function is said to exhibit dynamic scaling if it satisfies:

:<math>f(x,t)\sim t^\theta \varphi \left( \frac x {t^z} \right).</math>

Here the exponent <math>\theta</math> is fixed by the dimensional requirement <math>[f]=[t^\theta]</math>. The numerical value of <math>f/t^\theta</math> should remain invariant despite the unit of measurement of <math>t</math> is changed by some factor since <math>\varphi</math> is a dimensionless quantity. 

Many of these systems evolve in a self-similar fashion in the sense that data obtained from the snapshot at any fixed time is similar to the respective data taken from the snapshot of any earlier or later time. That is, the system is similar to itself at different times. The litmus test of such self-similarity is provided by the dynamic scaling.

== History ==
The term "dynamic scaling" as one of the essential concepts to describe the dynamics of [[critical phenomena]] seems to originate in the seminal paper of [[Pierre Hohenberg]] and [[Bertrand Halperin]] (1977), namely they suggested "[...] that the wave vector- and frequency dependent susceptibility of a ferromagnet near its Curie point may be expressed as a function independent of <math>|T-T_C|</math> provided that the length and frequency scales, as well as the magnetization and magnetic field, are rescaled by appropriate powers of <math>|T-T_C|</math>.<ref>{{cite journal|first1=Pierre Claude|last1=Hohenberg|first2=Bertrand Israel|last2=Halperin|title=Theory of dynamic critical phenomena|journal=Reviews of Modern Physics|volume=49|issue=3|pages=435–479|date=1 July 1977|doi=10.1103/RevModPhys.49.435|bibcode=1977RvMP...49..435H |s2cid=122636335 |url=https://link.aps.org/doi/10.1103/RevModPhys.49.435}}."</ref>

Later [[Tamás Vicsek]] and [[Fereydoon Family]] proposed the idea of dynamic scaling in the context of diffusion-limited aggregation ([[Diffusion-limited aggregation|DLA]]) of clusters in two dimensions.<ref name=Vicsek1984 /> The form of their proposal for dynamic scaling was:

:<math>f(x,t)\sim t^{-w}x^{-\tau} \varphi \left( \frac x {t^z} \right),</math>

where the exponents satisfy the following relation:

:<math>w=(2-\tau)z.</math>

== Test ==
In such systems we can define a certain time-dependent [[stochastic variable]] <math>x</math>. We are interested in computing the probability distribution of <math>x</math> at various instants of time i.e. <math>f(x,t)</math>. The numerical value of <math>f</math> and the typical or mean value of <math>x</math> generally changes over time. The question is: what happens to the corresponding dimensionless variables? If the numerical values of the dimensional quantities change, but corresponding dimensionless quantities remain invariant then we can argue that snapshots of the system at different times are similar. When this happens we say that the system is self-similar. 

One way of verifying dynamic scaling is to plot dimensionless variables <math>f/t^\theta</math> as a function of <math>x/t^z</math> of the data extracted at various different time. Then if all the plots of <math>f</math> vs <math>x</math> obtained at different times collapse onto a single universal curve then it is said that the systems at different time are similar and it obeys dynamic scaling. The idea of data collapse is deeply rooted to the [[Buckingham π theorem|Buckingham Pi theorem]].<ref>{{cite book | last=Barenblatt | first=G. I. | title=Scaling, self-similarity, and intermediate asymptotics | publisher=Cambridge University Press | publication-place=Cambridge New York | year=1996 | isbn=978-0-521-43522-2 | oclc=33946899 }}</ref> Essentially such systems can be termed as temporal self-similarity since the same system is similar at different times.

== Examples ==
Many phenomena investigated by physicists are not static but evolve probabilistically with time (i.e. [[Stochastic process]]). The universe itself is perhaps one of the best examples. It has been expanding ever since the [[Big Bang]]. Similarly, growth of [[Network science|networks]] like the [[Internet]] are also ever growing systems. Another example is [[polymer degradation]]<ref>{{cite journal | last1=Ziff | first1=R M | last2=McGrady | first2=E D | title=The kinetics of cluster fragmentation and depolymerisation | journal=Journal of Physics A: Mathematical and General | publisher=IOP Publishing | volume=18 | issue=15 | date=1985-10-21 | issn=0305-4470 | doi=10.1088/0305-4470/18/15/026 | pages=3027–3037| bibcode=1985JPhA...18.3027Z | hdl=2027.42/48803 | hdl-access=free }}</ref> where degradation does not occur in a blink of an eye but rather over quite a long time. Spread of biological and [[computer virus]]es too does not happen over night. 

Many other seemingly disparate systems which are found to exhibit dynamic scaling. For example:

* kinetics of aggregation described by [[Smoluchowski coagulation equation]],<ref>{{cite journal | last1=van Dongen | first1=P. G. J. | last2=Ernst | first2=M. H. | title=Dynamic Scaling in the Kinetics of Clustering | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=54 | issue=13 | date=1985-04-01 | issn=0031-9007 | doi=10.1103/physrevlett.54.1396 | pages=1396–1399| pmid=10031021 | bibcode=1985PhRvL..54.1396V }}</ref> <ref>{{Cite journal | doi = 10.1007/BF02186868| title = Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel| journal = Journal of Statistical Physics| volume = 75| issue = 3| pages = 389–407| year = 1994| last1 = Kreer | first1 = Markus | last2=Penrose | first2 = Oliver | bibcode = 1994JSP....75..389K| s2cid = 17392921}}</ref><ref>{{cite journal | last1=Hassan | first1=M. K. | last2=Hassan | first2=M. Z. | title=Emergence of fractal behavior in condensation-driven aggregation | journal=Physical Review E | volume=79 | issue=2 | date=2009-02-19 | issn=1539-3755 | doi=10.1103/physreve.79.021406 | page=021406| pmid=19391746 | arxiv=0901.2761 | bibcode=2009PhRvE..79b1406H | s2cid=26023004 }}</ref><ref>{{cite journal | last1=Hassan | first1=M. K. | last2=Hassan | first2=M. Z. | title=Condensation-driven aggregation in one dimension | journal=Physical Review E | publisher=American Physical Society (APS) | volume=77 | issue=6 | date=2008-06-13 | issn=1539-3755 | doi=10.1103/physreve.77.061404 | page=061404| pmid=18643263 | arxiv=0806.4872 | bibcode=2008PhRvE..77f1404H | s2cid=32261771 }}</ref><ref>{{cite journal | last1=Hassan | first1=Md. Kamrul | last2=Hassan | first2=Md. Zahedul | last3=Islam | first3=Nabila | title=Emergence of fractals in aggregation with stochastic self-replication | journal=Physical Review E | volume=88 | issue=4 | date=2013-10-24 | issn=1539-3755 | doi=10.1103/physreve.88.042137 | page=042137| pmid=24229145 | arxiv=1307.7804 | bibcode=2013PhRvE..88d2137H | s2cid=30562144 }}</ref> 
* [[Complex network|complex networks]] described by [[Barabasi–Albert model]],<ref>{{cite journal | last1=Hassan | first1=M Kamrul | last2=Hassan | first2=M Zahedul | last3=Pavel | first3=Neeaj I | title=Dynamic scaling, data-collapse and self-similarity in Barabási–Albert networks | journal=Journal of Physics A: Mathematical and Theoretical | publisher=IOP Publishing | volume=44 | issue=17 | date=2011-04-04 | issn=1751-8113 | doi=10.1088/1751-8113/44/17/175101 | page=175101| arxiv=1101.4730 | bibcode=2011JPhA...44q5101K | s2cid=15700641 }}</ref> 
* the kinetic and stochastic [[Cantor set]],<ref>{{cite journal | last1=Hassan | first1=M.K. | last2=Pavel | first2=N.I. | last3=Pandit | first3=R.K. | last4=Kurths | first4=J. | title=Dyadic Cantor set and its kinetic and stochastic counterpart | journal=Chaos, Solitons & Fractals | publisher=Elsevier BV | volume=60 | year=2014 | issn=0960-0779 | doi=10.1016/j.chaos.2013.12.010 | pages=31–39| arxiv=1401.0249 | bibcode=2014CSF....60...31H | s2cid=14494072 }}</ref>  
* the [[Surface growth|growth model]] within the [[Kardar–Parisi–Zhang equation|Kardar–Parisi–Zhang]] (KPZ) universality class; one find that the width of the surface <math>W(L,t)</math> exhibits dynamic scaling.<ref>{{cite journal|first1=Mehran|last1=Kardar|first2=Giorgio|last2=Parisi|first3=Yi-Cheng|last3=Zhang|title=Dynamic Scaling of Growing Interfaces|journal=Physical Review Letters|volume=56|issue=9|pages=889–892|date=3 March 1986|doi=10.1103/PhysRevLett.56.889|pmid=10033312|bibcode=1986PhRvL..56..889K|url=https://zenodo.org/record/1233849}}.</ref><ref>{{cite journal | last=D'souza | first=Raissa M. | title=Anomalies in Simulations of Nearest Neighbor Ballistic Deposition | journal=International Journal of Modern Physics C | publisher=World Scientific Pub Co Pte Lt | volume=08 | issue=4 | year=1997 | issn=0129-1831 | doi=10.1142/s0129183197000813 | pages=941–951| bibcode=1997IJMPC...8..941D }}</ref> 
* the area size distribution of the blocks of [[weighted planar stochastic lattice (WPSL)]] also exhibits dynamic scaling.{{Citation needed|date=June 2019}}
* the marginal probabilities of fractional Poisson processes exhibits dynamic scaling. <ref>{{cite journal | last=Kreer | first=Markus | title=An elementary proof for dynamical scaling for certain fractional non-homogeneous Poisson processes | journal=Statistics & Probability Letters | publisher=Elsevier B.V. | volume=182 | issue=61 | date=2022 | issn=0167-7152 | doi=10.1016/j.spl.2021.109296 | page=109296| arxiv=2103.07381 | s2cid=232222701 }}</ref>

==References==
{{Reflist}}

[[Category:Physical phenomena]]
[[Category:Stochastic models]]