20:
627:
is the distance to the nearest boundary. Because the scale of motions in the atmosphere is not limited, as in rivers or the subsurface, a plume continues to experience larger mixing motions as it increases in size, which also increases its diffusivity, resulting in super-diffusion.
636:
The types of anomalous diffusion given above allows one to measure the type, but how does anomalous diffusion arise? There are many possible ways to mathematically define a stochastic process which then has the right kind of power law. Some models are given here.
381:< 1: subdiffusion. This can happen due to crowding or walls. For example, a random walker in a crowded room, or in a maze, is able to move as usual for small random steps, but cannot take large random steps, creating subdiffusion. This appears for example in
218:
It has been found that equations describing normal diffusion are not capable of characterizing some complex diffusion processes, for instance, diffusion process in inhomogeneous or heterogeneous medium, e.g. porous media.
325:
201:
124:
63:
435:
556:
506:
516:
demonstrated that the atmosphere exhibits super-diffusion. In a bounded system, the mixing length (which determines the scale of dominant mixing motions) is given by the
352:
605:
478:
583:
372:
625:
1181:
Regner, Benjamin M.; Vučinić, Dejan; Domnisoru, Cristina; Bartol, Thomas M.; Hetzer, Martin W.; Tartakovsky, Daniel M.; Sejnowski, Terrence J. (2013).
1012:
876:
Sagi, Yoav; Brook, Miri; Almog, Ido; Davidson, Nir (2012). "Observation of
Anomalous Diffusion and Fractional Self-Similarity in One Dimension".
648:(fBm), and diffusion in disordered media. Currently the most studied types of anomalous diffusion processes are those involving the following
1774:
Toivonen, Matti S.; Onelli, Olimpia D.; Jacucci, Gianni; Lovikka, Ville; Rojas, Orlando J.; Ikkala, Olli; Vignolini, Silvia (13 March 2018).
265:
2202:"Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking"
1825:"Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking"
226:
Examples of anomalous diffusion in nature have been observed in ultra-cold atoms, harmonic spring-mass systems, scalar mixing in the
798:"Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen"
1705:
Masoliver, Jaume; Montero, Miquel; Weiss, George H. (2003). "Continuous-time random-walk model for financial distributions".
1445:"Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions"
2077:
von Kameke, A.; et al. (2010). "Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow".
149:
945:
Saporta-Katz, Ori; Efrati, Efi (2019). "Self-Driven
Fractional Rotational Diffusion of the Harmonic Three-Mass System".
2272:
1067:
Bronshtein, Irena; Israel, Yonatan; Kepten, Eldad; Mai, Sabina; Shav-Tal, Yaron; Barkai, Eli; Garini, Yuval (2009).
87:
26:
765:
707:
have shown that the motion of molecules in live cells often show a type of anomalous diffusion that breaks the
2314:
2309:
2151:"A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media"
1494:"Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors"
385:
diffusion within cells, or diffusion through porous media. Subdiffusion has been proposed as a measure of
2329:
1378:"A Dual-Permeability Approach to Study Anomalous Moisture Transport Properties of Cement-Based Materials"
1116:"Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking"
720:
677:
641:
1068:
657:
645:
408:
523:
442:
139:
81:
1936:
517:
485:
1248:"Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells"
716:
330:
2252:
588:
454:
2048:
Bouchaud, Jean-Philippe; Georges, Antoine (1990). "Anomalous diffusion in disordered media".
753:
212:
1638:
2213:
2162:
2086:
2057:
2010:
1948:
1893:
1836:
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1389:
1332:
1269:
1194:
1127:
1080:
1031:
964:
895:
850:
809:
561:
513:
451:= 2: ballistic motion. The prototypical example is a particle moving at constant velocity:
357:
1881:
1247:
8:
1960:
712:
227:
220:
135:
2217:
2166:
2090:
2061:
2014:
1952:
1937:"A review of progress in single particle tracking: from methods to biophysical insights"
1897:
1840:
1791:
1728:
1675:
1613:
1509:
1469:
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1444:
1393:
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1035:
968:
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813:
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2183:
2150:
2110:
2031:
1998:
1980:
1917:
1756:
1714:
1687:
1578:
1558:
1539:
1425:
1353:
1321:"A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model"
1320:
1301:
1259:
1223:
1182:
1158:
1115:
1049:
1021:
988:
954:
927:
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708:
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251:
1775:
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2069:
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1972:
1964:
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1805:
1748:
1740:
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1582:
1531:
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1429:
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1293:
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1228:
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1163:
1145:
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992:
980:
919:
911:
2290:
2174:
2149:
Sun, HongGuang; Meerschaert, Mark M.; Zhang, Yong; Zhu, Jianting; Chen, Wen (2013).
2114:
1984:
1543:
1053:
931:
374:
is the elapsed time. The classes of anomalous diffusions are classified as follows:
2260:
2221:
2178:
2170:
2135:
2094:
2065:
2026:
2018:
1956:
1901:
1844:
1795:
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1732:
1679:
1617:
1602:
Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences
1570:
1521:
1513:
1464:
1407:
1397:
1348:
1340:
1281:
1277:
1218:
1202:
1153:
1135:
1092:
1088:
1039:
1006:
Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017).
976:
972:
907:
903:
858:
817:
438:
1776:"Anomalous-Diffusion-Assisted Brightness in White Cellulose Nanofibril Membranes"
684:
653:
400:
131:
127:
2140:
2123:
2022:
1662:
Berkowicz, Ruwim (1984). "Spectral methods for atmospheric diffusion modeling".
1344:
1114:
Weigel, Aubrey V.; Simon, Blair; Tamkun, Michael M.; Krapf, Diego (2011-04-19).
838:
2264:
2098:
1736:
1517:
1412:
1402:
1377:
839:"Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"
759:
696:
386:
1999:"Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells"
1443:
Jeon, Jae-Hyung; Leijnse, Natascha; Oddershede, Lene B; Metzler, Ralf (2013).
1206:
1069:"Transient anomalous diffusion of telomeres in the nucleus of mammalian cells"
732:
2323:
2235:
1968:
1913:
1858:
1744:
1478:
1421:
1289:
1214:
1149:
915:
862:
822:
797:
672:
243:
1140:
2282:
2243:
2192:
2106:
2040:
1997:
Weiss, Matthias; Elsner, Markus; Kartberg, Fredrik; Nilsson, Tommy (2004).
1976:
1866:
1809:
1800:
1752:
1622:
1597:
1535:
1362:
1297:
1232:
1167:
1100:
1044:
1007:
984:
923:
704:
262:
Unlike typical diffusion, anomalous diffusion is described by a power law,
235:
19:
2315:
Anomalous interface shift kinetics (Computer simulations and
Experiments)
2259:, Current Topics in Membranes, vol. 75, Elsevier, pp. 167–207,
2200:
Metzler, Ralf; Jeon, Jae-Hyung; Cherstvy, Andrey G.; Barkai, Eli (2014).
1823:
Metzler, Ralf; Jeon, Jae-Hyung; Cherstvy, Andrey G.; Barkai, Eli (2014).
1719:
744:
738:
700:
668:
239:
223:
were introduced in order to characterize anomalous diffusion phenomena.
2226:
2201:
1849:
1824:
1683:
1526:
692:
688:
320:{\displaystyle \langle r^{2}(\tau )\rangle =K_{\alpha }\tau ^{\alpha }}
77:
1905:
1574:
1246:
Sabri, Adal; Xu, Xinran; Krapf, Diego; Weiss, Matthias (2020-07-28).
1008:"Scaling laws of passive-scalar diffusion in the interstellar medium"
390:
247:
231:
143:
73:
2124:"Anomalous diffusion modeling by fractal and fractional derivatives"
1492:
Bruno, L.; Levi, V.; Brunstein, M.; DespĂłsito, M. A. (2009-07-17).
1264:
1026:
959:
2253:"Mechanisms Underlying Anomalous Diffusion in the Plasma Membrane"
890:
711:. This type of motion require novel formalisms for the underlying
775:
664:
382:
2122:
Chen, Wen; Sun, HongGuang; Zhang, Xiaodi; Korosak, Dean (2010).
762: – A measure of the long-range dependence of a time series
250:, moisture transport in cement-based materials, and worm-like
1773:
1442:
1180:
1598:"Atmospheric Diffusion Shown on a Distance-Neighbour Graph"
1491:
691:
importance. Of particular interest, works by the groups of
1996:
687:
where the mechanism behind anomalous diffusion has direct
508:: hyperballistic. It has been observed in optical systems.
2199:
1822:
1066:
1643:. New Hampshire: John Wiley & Sons. pp. 145–150
16:
Diffusion process with a non-linear relationship to time
1005:
354:
is the so-called generalized diffusion coefficient and
2148:
1183:"Anomalous Diffusion of Single Particles in Cytoplasm"
1113:
875:
640:
These are long range correlations between the signals
437:: superdiffusion. Superdiffusion can be the result of
2310:
Boltzmann's transformation, Parabolic law (animation)
1704:
747: – Filtration of fluids through porous materials
741: – Process forming a path from many random steps
613:
591:
564:
526:
488:
457:
411:
360:
333:
268:
196:{\displaystyle \langle r^{2}(\tau )\rangle =2dD\tau }
152:
90:
29:
1698:
2121:
1935:Manzo, Carlo; Garcia-Parajo, Maria F (2015-12-01).
1556:
756: – Phenomenon in linguistics and data analysis
1934:
944:
778: – Infinitely detailed mathematical structure
735: – Random walk with heavy-tailed step lengths
619:
599:
577:
550:
500:
472:
429:
366:
346:
319:
195:
126:, and time. This behavior is in stark contrast to
118:
57:
1245:
1013:Monthly Notices of the Royal Astronomical Society
257:
2321:
1174:
836:
631:
2047:
1436:
1120:Proceedings of the National Academy of Sciences
1636:
1060:
938:
130:, the typical diffusion process described by
1879:
869:
291:
269:
175:
153:
119:{\displaystyle \langle r^{2}(\tau )\rangle }
113:
91:
58:{\displaystyle \langle r^{2}(\tau )\rangle }
52:
30:
2128:Computers and Mathematics with Applications
2076:
1880:Krapf, Diego; Metzler, Ralf (2019-09-01).
1595:
1376:Zhang, Zhidong; Angst, Ueli (2020-10-01).
1375:
65:for different types of anomalous diffusion
2225:
2182:
2139:
2030:
1848:
1799:
1718:
1661:
1621:
1525:
1468:
1411:
1401:
1352:
1263:
1222:
1157:
1139:
1043:
1025:
958:
889:
821:
683:These processes have growing interest in
1882:"Strange interfacial molecular dynamics"
795:
18:
1319:Saxton, Michael J. (15 February 2007).
2322:
1318:
2250:
1637:Cushman-Roisin, Benoit (March 2014).
207:being the number of dimensions and
13:
1816:
1596:Richardson, L. F. (1 April 1926).
14:
2341:
2303:
512:In 1926, using weather balloons,
441:processes or due to jumps with a
430:{\displaystyle 1<\alpha <2}
607:is the Von Kármán constant, and
551:{\displaystyle l_{m}={\kappa }z}
2175:10.1016/j.advwatres.2012.11.005
1928:
1873:
1767:
1655:
1630:
1589:
1550:
1485:
1369:
1312:
1961:10.1088/0034-4885/78/12/124601
1941:Reports on Progress in Physics
1282:10.1103/PhysRevLett.125.058101
1239:
1107:
1093:10.1103/PhysRevLett.103.018102
999:
977:10.1103/PhysRevLett.122.024102
908:10.1103/PhysRevLett.108.093002
830:
789:
766:Detrended fluctuation analysis
288:
282:
258:Classes of anomalous diffusion
221:Fractional diffusion equations
172:
166:
110:
104:
49:
43:
1:
1640:Environmental Fluid Mechanics
1470:10.1088/1367-2630/15/4/045011
837:von Smoluchowski, M. (1906).
782:
632:Models of anomalous diffusion
2070:10.1016/0370-1573(90)90099-N
1561:(2012). "Beyond ballistic".
678:Continuous time random walks
642:continuous-time random walks
501:{\displaystyle \alpha >2}
246:, colloidal particle in the
7:
2155:Advances in Water Resources
2141:10.1016/j.camwa.2009.08.020
2023:10.1529/biophysj.104.044263
1345:10.1529/biophysj.106.092619
726:
347:{\displaystyle K_{\alpha }}
10:
2346:
2265:10.1016/bs.ctm.2015.03.002
2099:10.1103/physreve.81.066211
1737:10.1103/PhysRevE.67.021112
1664:Boundary-Layer Meteorology
1518:10.1103/PhysRevE.80.011912
1403:10.1007/s11242-020-01469-y
660:and scaled Brownian motion
658:fractional Brownian motion
646:fractional Brownian motion
520:according to the equation
23:Mean squared displacement
1382:Transport in Porous Media
1207:10.1016/j.bpj.2013.01.049
772:) – Statistical term
715:because approaches using
600:{\displaystyle {\kappa }}
439:active cellular transport
82:mean squared displacement
80:relationship between the
863:10.1002/andp.19063261405
823:10.1002/andp.19053220806
473:{\displaystyle r=v\tau }
1252:Physical Review Letters
1141:10.1073/pnas.1016325108
1073:Physical Review Letters
947:Physical Review Letters
878:Physical Review Letters
721:Wiener–Khinchin theorem
717:microcanonical ensemble
695:, Maria Garcia Parajo,
443:heavy-tail distribution
2206:Phys. Chem. Chem. Phys
1829:Phys. Chem. Chem. Phys
1801:10.1002/adma.201704050
1623:10.1098/rspa.1926.0043
1449:New Journal of Physics
750:Long term correlations
621:
601:
585:is the mixing length,
579:
552:
502:
474:
431:
368:
348:
321:
197:
120:
66:
59:
2251:Krapf, Diego (2015),
796:Einstein, A. (1905).
754:Long-range dependence
622:
602:
580:
578:{\displaystyle l_{m}}
553:
503:
475:
432:
369:
367:{\displaystyle \tau }
349:
322:
213:diffusion coefficient
198:
121:
60:
22:
1045:10.1093/mnras/stx261
611:
589:
562:
524:
514:Lewis Fry Richardson
486:
455:
409:
358:
331:
266:
150:
88:
27:
2218:2014PCCP...1624128M
2212:(44): 24128–24164.
2167:2013AdWR...52..292S
2091:2010PhRvE..81f6211V
2062:1990PhR...195..127B
2015:2004BpJ....87.3518W
2003:Biophysical Journal
1953:2015RPPh...78l4601M
1898:2019PhT....72i..48K
1841:2014PCCP...1624128M
1835:(44): 24128–24164.
1792:2018AdM....3004050T
1729:2003PhRvE..67b1112M
1676:1984BoLMe..30..201B
1614:1926RSPSA.110..709R
1559:Morandotti, Roberto
1510:2009PhRvE..80a1912B
1461:2013NJPh...15d5011J
1413:20.500.11850/438735
1394:2020TPMed.135...59Z
1337:2007BpJ....92.1178S
1325:Biophysical Journal
1274:2020PhRvL.125e8101S
1199:2013BpJ...104.1652R
1187:Biophysical Journal
1132:2011PNAS..108.6438W
1085:2009PhRvL.103a8102B
1036:2017MNRAS.467.2421C
969:2019PhRvL.122b4102S
900:2012PhRvL.108i3002S
855:1906AnP...326..756V
814:1905AnP...322..549E
713:statistical physics
652:Generalizations of
518:Von Kármán constant
228:interstellar medium
70:Anomalous diffusion
2330:Physical chemistry
2227:10.1039/c4cp03465a
1850:10.1039/C4CP03465A
1780:Advanced Materials
1684:10.1007/BF00121955
1557:Peccianti, Marco;
843:Annalen der Physik
802:Annalen der Physik
709:ergodic hypothesis
617:
597:
575:
548:
498:
470:
427:
364:
344:
317:
252:micellar solutions
193:
116:
67:
55:
1906:10.1063/PT.3.4294
1707:Physical Review E
1575:10.1038/nphys2486
1498:Physical Review E
1126:(16): 6438–6443.
620:{\displaystyle z}
146:in time (namely,
2337:
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2298:
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2229:
2196:
2186:
2145:
2143:
2134:(5): 1754–1758.
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2073:
2056:(4–5): 127–293.
2044:
2034:
2009:(5): 3518–3524.
1989:
1988:
1932:
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1720:cond-mat/0210513
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1608:(756): 709–737.
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1331:(4): 1178–1191.
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1193:(8): 1652–1660.
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1020:(2): 2421–2429.
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389:crowding in the
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2050:Physics Reports
1993:
1992:
1933:
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1786:(16): 1704050.
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1703:
1699:
1660:
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1646:
1644:
1635:
1631:
1594:
1590:
1569:(12): 858–859.
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874:
870:
849:(14): 756–780.
835:
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769:
729:
685:cell biophysics
654:Brownian motion
634:
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401:Brownian motion
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244:plasma membrane
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128:Brownian motion
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76:process with a
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2304:External links
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1991:
1990:
1947:(12): 124601.
1927:
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1670:(1): 201–219.
1654:
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1588:
1563:Nature Physics
1549:
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868:
829:
808:(8): 549–560.
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760:Hurst exponent
757:
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742:
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697:Joseph Klafter
681:
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656:, such as the
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