651:
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
255:
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
473:
107:
765:
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
276:(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
688:
523:
210:
360:
317:
250:
723:
130:
874:
609:
763:
743:
649:
629:
574:
554:
230:
175:
383:
1427:
Hassan, M Kamrul; Hassan, M Zahedul; Pavel, Neeaj I (2011-04-04). "Dynamic scaling, data-collapse and self-similarity in Barabási–Albert networks".
33:
1358:
Hassan, Md. Kamrul; Hassan, Md. Zahedul; Islam, Nabila (2013-10-24). "Emergence of fractals in aggregation with stochastic self-replication".
319:
provided that the length and frequency scales, as well as the magnetization and magnetic field, are rescaled by appropriate powers of
880:
1177:
Kreer, Markus; Penrose, Oliver (1994). "Proof of dynamical scaling in
Smoluchowski's coagulation equation with constant kernel".
1049:
1640:
Kreer, Markus (2022). "An elementary proof for dynamical scaling for certain fractional non-homogeneous
Poisson processes".
1488:
Hassan, M.K.; Pavel, N.I.; Pandit, R.K.; Kurths, J. (2014). "Dyadic Cantor set and its kinetic and stochastic counterpart".
838:
770:. Essentially such systems can be termed as temporal self-similarity since the same system is similar at different times.
911:(1985). "Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model".
810:
798:
where degradation does not occur in a blink of an eye but rather over quite a long time. Spread of biological and
1220:
Hassan, M. K.; Hassan, M. Z. (2009-02-19). "Emergence of fractal behavior in condensation-driven aggregation".
374:
1699:
1704:
778:
Many phenomena investigated by physicists are not static but evolve probabilistically with time (i.e.
820:
1073:
Ziff, R M; McGrady, E D (1985-10-21). "The kinetics of cluster fragmentation and depolymerisation".
658:
484:
180:
1126:
van Dongen, P. G. J.; Ernst, M. H. (1985-04-01). "Dynamic
Scaling in the Kinetics of Clustering".
782:). The universe itself is perhaps one of the best examples. It has been expanding ever since the
1597:
D'souza, Raissa M. (1997). "Anomalies in
Simulations of Nearest Neighbor Ballistic Deposition".
805:
Many other seemingly disparate systems which are found to exhibit dynamic scaling. For example:
767:
1289:
Hassan, M. K.; Hassan, M. Z. (2008-06-13). "Condensation-driven aggregation in one dimension".
322:
279:
949:
Vicsek, Tamás; Family, Fereydoon (1984-05-07). "Dynamic
Scaling for Aggregation of Clusters".
235:
693:
115:
844:
579:
1606:
1562:
1507:
1446:
1377:
1308:
1239:
1186:
1135:
1082:
1005:
958:
920:
8:
1458:
795:
534:
1610:
1566:
1511:
1450:
1381:
1312:
1243:
1190:
1139:
1086:
1009:
962:
924:
264:
The term "dynamic scaling" as one of the essential concepts to describe the dynamics of
1675:
1649:
1531:
1497:
1470:
1436:
1409:
1367:
1340:
1298:
1271:
1229:
1202:
1021:
779:
748:
728:
634:
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539:
265:
215:
135:
1094:
373:
proposed the idea of dynamic scaling in the context of diffusion-limited aggregation (
1679:
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1401:
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1332:
1324:
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1255:
1159:
1151:
1108:
1055:
1045:
1025:
993:
974:
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the marginal probabilities of fractional
Poisson processes exhibits dynamic scaling.
377:) of clusters in two dimensions. The form of their proposal for dynamic scaling was:
273:
1535:
1474:
1413:
1344:
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1206:
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1143:
1098:
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1013:
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928:
468:{\displaystyle f(x,t)\sim t^{-w}x^{-\tau }\varphi \left({\frac {x}{t^{z}}}\right),}
370:
269:
816:
787:
24:
1519:
1147:
970:
366:
1574:
1389:
1320:
1251:
834:
799:
1663:
1618:
1017:
102:{\displaystyle f(x,t)\sim t^{\theta }\varphi \left({\frac {x}{t^{z}}}\right).}
1693:
1671:
1626:
1527:
1466:
1397:
1328:
1259:
1155:
1112:
978:
1059:
27:. In general a function is said to exhibit dynamic scaling if it satisfies:
1582:
1405:
1336:
1267:
1163:
725:
of the data extracted at various different time. Then if all the plots of
1103:
655:
One way of verifying dynamic scaling is to plot dimensionless variables
1198:
827:
908:
904:
1550:
652:
similar. When this happens we say that the system is self-similar.
1654:
992:
Hohenberg, Pierre Claude; Halperin, Bertrand Israel (1 July 1977).
791:
783:
1502:
1441:
1372:
1303:
1234:
23:) is a litmus test that shows whether an evolving system exhibits
1549:
Kardar, Mehran; Parisi, Giorgio; Zhang, Yi-Cheng (3 March 1986).
841:(KPZ) universality class; one find that the width of the surface
556:. We are interested in computing the probability distribution of
212:
should remain invariant despite the unit of measurement of
1487:
533:
In such systems we can define a certain time-dependent
1042:
Scaling, self-similarity, and intermediate asymptotics
847:
751:
731:
696:
661:
637:
617:
582:
562:
542:
487:
386:
325:
282:
238:
218:
183:
138:
118:
36:
478:
where the exponents satisfy the following relation:
256:
self-similarity is provided by the dynamic scaling.
1429:Journal of Physics A: Mathematical and Theoretical
1134:(13). American Physical Society (APS): 1396–1399.
1044:. Cambridge New York: Cambridge University Press.
957:(19). American Physical Society (APS): 1669–1672.
868:
794:are also ever growing systems. Another example is
757:
737:
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643:
623:
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568:
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467:
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311:
244:
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169:
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101:
1548:
1426:
1357:
991:
1691:
1125:
1605:(4). World Scientific Pub Co Pte Lt: 941–951.
1297:(6). American Physical Society (APS): 061404.
1075:Journal of Physics A: Mathematical and General
913:Journal of Physics A: Mathematical and General
1288:
1219:
1176:
948:
944:
942:
903:
879:the area size distribution of the blocks of
1596:
1072:
268:seems to originate in the seminal paper of
1039:
1653:
1599:International Journal of Modern Physics C
1501:
1440:
1371:
1302:
1233:
1102:
939:
881:weighted planar stochastic lattice (WPSL)
132:is fixed by the dimensional requirement
1551:"Dynamic Scaling of Growing Interfaces"
1692:
994:"Theory of dynamic critical phenomena"
1639:
809:kinetics of aggregation described by
1642:Statistics & Probability Letters
13:
14:
1716:
1081:(15). IOP Publishing: 3027–3037.
811:Smoluchowski coagulation equation
802:too does not happen over night.
631:and the typical or mean value of
576:at various instants of time i.e.
232:is changed by some factor since
1633:
1590:
1542:
1481:
1420:
1351:
1490:Chaos, Solitons & Fractals
1459:10.1088/1751-8113/44/17/175101
1435:(17). IOP Publishing: 175101.
1282:
1213:
1179:Journal of Statistical Physics
1170:
1119:
1066:
1033:
985:
897:
883:also exhibits dynamic scaling.
863:
851:
598:
586:
506:
494:
402:
390:
348:
327:
305:
284:
252:is a dimensionless quantity.
164:
151:
145:
139:
52:
40:
1:
1648:(61). Elsevier B.V.: 109296.
890:
683:{\displaystyle f/t^{\theta }}
518:{\displaystyle w=(2-\tau )z.}
205:{\displaystyle f/t^{\theta }}
7:
1520:10.1016/j.chaos.2013.12.010
1148:10.1103/physrevlett.54.1396
1095:10.1088/0305-4470/18/15/026
971:10.1103/physrevlett.52.1669
826:the kinetic and stochastic
773:
10:
1721:
1575:10.1103/PhysRevLett.56.889
1390:10.1103/physreve.88.042137
1321:10.1103/physreve.77.061404
1252:10.1103/physreve.79.021406
1040:Barenblatt, G. I. (1996).
933:10.1088/0305-4470/18/2/005
259:
1664:10.1016/j.spl.2021.109296
1619:10.1142/s0129183197000813
1018:10.1103/RevModPhys.49.435
998:Reviews of Modern Physics
876:exhibits dynamic scaling.
611:. The numerical value of
355:{\displaystyle |T-T_{C}|}
312:{\displaystyle |T-T_{C}|}
177:. The numerical value of
245:{\displaystyle \varphi }
1555:Physical Review Letters
1128:Physical Review Letters
951:Physical Review Letters
786:. Similarly, growth of
718:{\displaystyle x/t^{z}}
528:
125:{\displaystyle \theta }
1496:. Elsevier BV: 31–39.
870:
869:{\displaystyle W(L,t)}
759:
739:
719:
684:
645:
625:
605:
604:{\displaystyle f(x,t)}
570:
550:
519:
469:
356:
313:
246:
226:
206:
171:
126:
103:
871:
821:Barabasi–Albert model
768:Buckingham Pi theorem
760:
740:
720:
685:
646:
626:
606:
571:
551:
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470:
357:
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247:
227:
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127:
104:
21:Family–Vicsek scaling
845:
749:
729:
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635:
615:
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323:
280:
236:
216:
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34:
19:(sometimes known as
1611:1997IJMPC...8..941D
1567:1986PhRvL..56..889K
1512:2014CSF....60...31H
1451:2011JPhA...44q5101K
1382:2013PhRvE..88d2137H
1313:2008PhRvE..77f1404H
1244:2009PhRvE..79b1406H
1191:1994JSP....75..389K
1140:1985PhRvL..54.1396V
1087:1985JPhA...18.3027Z
1010:1977RvMP...49..435H
963:1984PhRvL..52.1669V
925:1985JPhA...18L..75F
839:Kardar–Parisi–Zhang
796:polymer degradation
535:stochastic variable
1700:Physical phenomena
1199:10.1007/BF02186868
866:
780:Stochastic process
755:
735:
715:
680:
641:
621:
601:
566:
546:
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266:critical phenomena
242:
222:
202:
167:
122:
112:Here the exponent
99:
1705:Stochastic models
1360:Physical Review E
1291:Physical Review E
1222:Physical Review E
1051:978-0-521-43522-2
758:{\displaystyle x}
738:{\displaystyle f}
690:as a function of
644:{\displaystyle x}
624:{\displaystyle f}
569:{\displaystyle x}
549:{\displaystyle x}
456:
274:Bertrand Halperin
225:{\displaystyle t}
170:{\displaystyle =}
90:
1712:
1684:
1683:
1657:
1637:
1631:
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1594:
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989:
983:
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817:complex networks
800:computer viruses
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371:Fereydoon Family
361:
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353:
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346:
345:
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318:
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270:Pierre Hohenberg
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1120:
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117:
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113:
84:
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75:
71:
62:
58:
35:
32:
31:
25:self-similarity
17:Dynamic scaling
12:
11:
5:
1718:
1708:
1707:
1702:
1686:
1685:
1632:
1589:
1561:(9): 889–892.
1541:
1480:
1419:
1350:
1281:
1212:
1185:(3): 389–407.
1169:
1118:
1065:
1050:
1032:
1004:(3): 435–479.
984:
938:
919:(2): L75–L81.
895:
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865:
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1403:
1399:
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1383:
1379:
1374:
1369:
1366:(4): 042137.
1365:
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1342:
1338:
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1330:
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1322:
1318:
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1273:
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1241:
1236:
1231:
1228:(2): 021406.
1227:
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1208:
1204:
1200:
1196:
1192:
1188:
1184:
1180:
1173:
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1104:2027.42/48803
1100:
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836:
832:
829:
825:
822:
819:described by
818:
815:
812:
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801:
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793:
789:
785:
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771:
769:
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653:
638:
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119:
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68:
63:
59:
55:
49:
46:
43:
37:
30:
29:
28:
26:
22:
18:
1645:
1641:
1635:
1602:
1598:
1592:
1558:
1554:
1544:
1493:
1489:
1483:
1432:
1428:
1422:
1363:
1359:
1353:
1294:
1290:
1284:
1225:
1221:
1215:
1182:
1178:
1172:
1131:
1127:
1121:
1078:
1074:
1068:
1041:
1035:
1001:
997:
987:
954:
950:
916:
912:
899:
835:growth model
804:
777:
654:
532:
477:
367:Tamás Vicsek
364:
263:
254:
111:
20:
16:
15:
837:within the
1694:Categories
1655:2103.07381
909:Vicsek, T.
905:Family, F.
891:References
828:Cantor set
1680:232222701
1672:0167-7152
1627:0129-1831
1528:0960-0779
1503:1401.0249
1467:1751-8113
1442:1101.4730
1398:1539-3755
1373:1307.7804
1329:1539-3755
1304:0806.4872
1260:1539-3755
1235:0901.2761
1156:0031-9007
1113:0305-4470
1026:122636335
979:0031-9007
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