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Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), $\langle r^{2}(\tau )\rangle$ , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process described by Einstein and Smoluchowski, where the MSD is linear in time (namely, $\langle r^{2}(\tau )\rangle =2dD\tau$ with d being the number of dimensions and D the diffusion coefficient).^[1]^[2]

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. Fractional diffusion equations were introduced in order to characterize anomalous diffusion phenomena.

Examples of anomalous diffusion in nature have been observed in ultra-cold atoms,^[3] harmonic spring-mass systems,^[4] scalar mixing in the interstellar medium, ^[5] telomeres in the nucleus of cells,^[6] ion channels in the plasma membrane,^[7] colloidal particle in the cytoplasm,^[8]^[9]^[10] moisture transport in cement-based materials,^[11] and worm-like micellar solutions.^[12]

Classes of anomalous diffusion

Unlike typical diffusion, anomalous diffusion is described by a power law, $\langle r^{2}(\tau )\rangle =K_{\alpha }\tau ^{\alpha }$ where $K_{\alpha }$ is the so-called generalized diffusion coefficient and $\tau$ is the elapsed time. The classes of anomalous diffusions are classified as follows:

α < 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 protein diffusion within cells, or diffusion through porous media. Subdiffusion has been proposed as a measure of macromolecular crowding in the cytoplasm.
α = 1: Brownian motion.
$1<\alpha <2$ : superdiffusion. Superdiffusion can be the result of active cellular transport processes or due to jumps with a heavy-tail distribution.^[13]
α = 2: ballistic motion. The prototypical example is a particle moving at constant velocity: $r=v\tau$ .
$\alpha >2$ : hyperballistic. It has been observed in optical systems.^[14]

In 1926, using weather balloons, Lewis Fry Richardson demonstrated that the atmosphere exhibits super-diffusion.^[15] In a bounded system, the mixing length (which determines the scale of dominant mixing motions) is given by the Von Kármán constant according to the equation $l_{m}={\kappa }z$ , where $l_{m}$ is the mixing length, ${\kappa }$ is the Von Kármán constant, and $z$ is the distance to the nearest boundary.^[16] 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.^[17]

Models of anomalous diffusion

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.

These are long range correlations between the signals continuous-time random walks (CTRW)^[18] and fractional Brownian motion (fBm), and diffusion in disordered media.^[19] Currently the most studied types of anomalous diffusion processes are those involving the following

Generalizations of Brownian motion, such as the fractional Brownian motion and scaled Brownian motion
Diffusion in fractals and percolation in porous media
Continuous time random walks

These processes have growing interest in cell biophysics where the mechanism behind anomalous diffusion has direct physiological importance. Of particular interest, works by the groups of Eli Barkai, Maria Garcia Parajo, Joseph Klafter, Diego Krapf, and Ralf Metzler have shown that the motion of molecules in live cells often show a type of anomalous diffusion that breaks the ergodic hypothesis.^[20]^[21]^[22] This type of motion require novel formalisms for the underlying statistical physics because approaches using microcanonical ensemble and Wiener–Khinchin theorem break down.

References

^ Einstein, A. (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik (in German). 322 (8): 549–560. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806.
^ von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen". Annalen der Physik (in German). 326 (14): 756–780. Bibcode:1906AnP...326..756V. doi:10.1002/andp.19063261405.
^ Sagi, Yoav; Brook, Miri; Almog, Ido; Davidson, Nir (2012). "Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension". Physical Review Letters. 108 (9): 093002. arXiv:1109.1503. Bibcode:2012PhRvL.108i3002S. doi:10.1103/PhysRevLett.108.093002. ISSN 0031-9007. PMID 22463630. S2CID 24674876.
^ Saporta-Katz, Ori; Efrati, Efi (2019). "Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System". Physical Review Letters. 122 (2): 024102. arXiv:1706.09868. Bibcode:2019PhRvL.122b4102S. doi:10.1103/PhysRevLett.122.024102. PMID 30720293. S2CID 119240381.
^ Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261. S2CID 20203131.
^ Bronshtein, Irena; Israel, Yonatan; Kepten, Eldad; Mai, Sabina; Shav-Tal, Yaron; Barkai, Eli; Garini, Yuval (2009). "Transient anomalous diffusion of telomeres in the nucleus of mammalian cells". Physical Review Letters. 103 (1): 018102. Bibcode:2009PhRvL.103a8102B. doi:10.1103/PhysRevLett.103.018102. PMID 19659180.
^ Weigel, Aubrey V.; Simon, Blair; Tamkun, Michael M.; Krapf, Diego (2011-04-19). "Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking". Proceedings of the National Academy of Sciences. 108 (16): 6438–6443. Bibcode:2011PNAS..108.6438W. doi:10.1073/pnas.1016325108. ISSN 0027-8424. PMC 3081000. PMID 21464280.
^ Regner, Benjamin M.; Vučinić, Dejan; Domnisoru, Cristina; Bartol, Thomas M.; Hetzer, Martin W.; Tartakovsky, Daniel M.; Sejnowski, Terrence J. (2013). "Anomalous Diffusion of Single Particles in Cytoplasm". Biophysical Journal. 104 (8): 1652–1660. Bibcode:2013BpJ...104.1652R. doi:10.1016/j.bpj.2013.01.049. ISSN 0006-3495. PMC 3627875. PMID 23601312.
^ Sabri, Adal; Xu, Xinran; Krapf, Diego; Weiss, Matthias (2020-07-28). "Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells". Physical Review Letters. 125 (5): 058101. arXiv:1910.00102. Bibcode:2020PhRvL.125e8101S. doi:10.1103/PhysRevLett.125.058101. ISSN 0031-9007. PMID 32794890. S2CID 203610681.
^ Saxton, Michael J. (15 February 2007). "A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model". Biophysical Journal. 92 (4): 1178–1191. Bibcode:2007BpJ....92.1178S. doi:10.1529/biophysj.106.092619. PMC 1783867. PMID 17142285.
^ Zhang, Zhidong; Angst, Ueli (2020-10-01). "A Dual-Permeability Approach to Study Anomalous Moisture Transport Properties of Cement-Based Materials". Transport in Porous Media. 135 (1): 59–78. Bibcode:2020TPMed.135...59Z. doi:10.1007/s11242-020-01469-y. hdl:20.500.11850/438735. ISSN 1573-1634. S2CID 221495131.
^ Jeon, Jae-Hyung; Leijnse, Natascha; Oddershede, Lene B; Metzler, Ralf (2013). "Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions". New Journal of Physics. 15 (4): 045011. Bibcode:2013NJPh...15d5011J. doi:10.1088/1367-2630/15/4/045011. ISSN 1367-2630.
^ Bruno, L.; Levi, V.; Brunstein, M.; Despósito, M. A. (2009-07-17). "Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors". Physical Review E. 80 (1): 011912. Bibcode:2009PhRvE..80a1912B. doi:10.1103/PhysRevE.80.011912. hdl:11336/60415. PMID 19658734. S2CID 15216911.
^ Peccianti, Marco; Morandotti, Roberto (2012). "Beyond ballistic". Nature Physics. 8 (12): 858–859. doi:10.1038/nphys2486. S2CID 121404743.
^ Richardson, L. F. (1 April 1926). "Atmospheric Diffusion Shown on a Distance-Neighbour Graph". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 110 (756): 709–737. Bibcode:1926RSPSA.110..709R. doi:10.1098/rspa.1926.0043.
^ Cushman-Roisin, Benoit (March 2014). Environmental Fluid Mechanics. New Hampshire: John Wiley & Sons. pp. 145–150. Retrieved 28 April 2017.
^ Berkowicz, Ruwim (1984). "Spectral methods for atmospheric diffusion modeling". Boundary-Layer Meteorology. 30 (1): 201–219. Bibcode:1984BoLMe..30..201B. doi:10.1007/BF00121955. S2CID 121838208.
^ Masoliver, Jaume; Montero, Miquel; Weiss, George H. (2003). "Continuous-time random-walk model for financial distributions". Physical Review E. 67 (2): 021112. arXiv:cond-mat/0210513. Bibcode:2003PhRvE..67b1112M. doi:10.1103/PhysRevE.67.021112. ISSN 1063-651X. PMID 12636658. S2CID 2966272.
^ Toivonen, Matti S.; Onelli, Olimpia D.; Jacucci, Gianni; Lovikka, Ville; Rojas, Orlando J.; Ikkala, Olli; Vignolini, Silvia (13 March 2018). "Anomalous-Diffusion-Assisted Brightness in White Cellulose Nanofibril Membranes". Advanced Materials. 30 (16): 1704050. Bibcode:2018AdM....3004050T. doi:10.1002/adma.201704050. PMID 29532967.
^ Metzler, Ralf; Jeon, Jae-Hyung; Cherstvy, Andrey G.; Barkai, Eli (2014). "Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking". Phys. Chem. Chem. Phys. 16 (44): 24128–24164. Bibcode:2014PCCP...1624128M. doi:10.1039/C4CP03465A. ISSN 1463-9076. PMID 25297814.
^ Krapf, Diego; Metzler, Ralf (2019-09-01). "Strange interfacial molecular dynamics". Physics Today. 72 (9): 48–54. Bibcode:2019PhT....72i..48K. doi:10.1063/PT.3.4294. ISSN 0031-9228. S2CID 203336692.
^ Manzo, Carlo; Garcia-Parajo, Maria F (2015-12-01). "A review of progress in single particle tracking: from methods to biophysical insights". Reports on Progress in Physics. 78 (12): 124601. Bibcode:2015RPPh...78l4601M. doi:10.1088/0034-4885/78/12/124601. ISSN 0034-4885. PMID 26511974. S2CID 25691993.

Weiss, Matthias; Elsner, Markus; Kartberg, Fredrik; Nilsson, Tommy (2004). "Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells". Biophysical Journal. 87 (5): 3518–3524. Bibcode:2004BpJ....87.3518W. doi:10.1529/biophysj.104.044263. PMC 1304817. PMID 15339818.
Bouchaud, Jean-Philippe; Georges, Antoine (1990). "Anomalous diffusion in disordered media". Physics Reports. 195 (4–5): 127–293. Bibcode:1990PhR...195..127B. doi:10.1016/0370-1573(90)90099-N.
von Kameke, A.; et al. (2010). "Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow". Phys. Rev. E. 81 (6): 066211. Bibcode:2010PhRvE..81f6211V. doi:10.1103/physreve.81.066211. PMID 20866505. S2CID 23202701.
Chen, Wen; Sun, HongGuang; Zhang, Xiaodi; Korosak, Dean (2010). "Anomalous diffusion modeling by fractal and fractional derivatives". Computers and Mathematics with Applications. 59 (5): 1754–1758. doi:10.1016/j.camwa.2009.08.020.
Sun, HongGuang; Meerschaert, Mark M.; Zhang, Yong; Zhu, Jianting; Chen, Wen (2013). "A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media". Advances in Water Resources. 52: 292–295. Bibcode:2013AdWR...52..292S. doi:10.1016/j.advwatres.2012.11.005. PMC 3686513. PMID 23794783.
Metzler, Ralf; Jeon, Jae-Hyung; Cherstvy, Andrey G.; Barkai, Eli (2014). "Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking". Phys. Chem. Chem. Phys. 16 (44): 24128–24164. Bibcode:2014PCCP...1624128M. doi:10.1039/c4cp03465a. ISSN 1463-9076. PMID 25297814.
Krapf, Diego (2015), "Mechanisms Underlying Anomalous Diffusion in the Plasma Membrane", Lipid Domains, Current Topics in Membranes, vol. 75, Elsevier, pp. 167–207, doi:10.1016/bs.ctm.2015.03.002, ISBN 9780128032954, PMID 26015283, S2CID 34712482, retrieved 2018-08-13

External links

[1] Einstein, A. (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen". Annalen der Physik (in German). 322 (8): 549–560. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806.

[2] von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen". Annalen der Physik (in German). 326 (14): 756–780. Bibcode:1906AnP...326..756V. doi:10.1002/andp.19063261405.

[SagiBrook2012-3] Sagi, Yoav; Brook, Miri; Almog, Ido; Davidson, Nir (2012). "Observation of Anomalous Diffusion and Fractional Self-Similarity in One Dimension". Physical Review Letters. 108 (9): 093002. arXiv:1109.1503. Bibcode:2012PhRvL.108i3002S. doi:10.1103/PhysRevLett.108.093002. ISSN 0031-9007. PMID 22463630. S2CID 24674876.

[KatzEfrati2019-4] Saporta-Katz, Ori; Efrati, Efi (2019). "Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System". Physical Review Letters. 122 (2): 024102. arXiv:1706.09868. Bibcode:2019PhRvL.122b4102S. doi:10.1103/PhysRevLett.122.024102. PMID 30720293. S2CID 119240381.

[5] Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017). "Scaling laws of passive-scalar diffusion in the interstellar medium". Monthly Notices of the Royal Astronomical Society. 467 (2): 2421–2429. arXiv:1610.06590. Bibcode:2017MNRAS.467.2421C. doi:10.1093/mnras/stx261. S2CID 20203131.

[Bronshtein2009-6] Bronshtein, Irena; Israel, Yonatan; Kepten, Eldad; Mai, Sabina; Shav-Tal, Yaron; Barkai, Eli; Garini, Yuval (2009). "Transient anomalous diffusion of telomeres in the nucleus of mammalian cells". Physical Review Letters. 103 (1): 018102. Bibcode:2009PhRvL.103a8102B. doi:10.1103/PhysRevLett.103.018102. PMID 19659180.

[7] Weigel, Aubrey V.; Simon, Blair; Tamkun, Michael M.; Krapf, Diego (2011-04-19). "Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking". Proceedings of the National Academy of Sciences. 108 (16): 6438–6443. Bibcode:2011PNAS..108.6438W. doi:10.1073/pnas.1016325108. ISSN 0027-8424. PMC 3081000. PMID 21464280.

[RegnerVučinić2013-8] Regner, Benjamin M.; Vučinić, Dejan; Domnisoru, Cristina; Bartol, Thomas M.; Hetzer, Martin W.; Tartakovsky, Daniel M.; Sejnowski, Terrence J. (2013). "Anomalous Diffusion of Single Particles in Cytoplasm". Biophysical Journal. 104 (8): 1652–1660. Bibcode:2013BpJ...104.1652R. doi:10.1016/j.bpj.2013.01.049. ISSN 0006-3495. PMC 3627875. PMID 23601312.

[9] Sabri, Adal; Xu, Xinran; Krapf, Diego; Weiss, Matthias (2020-07-28). "Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells". Physical Review Letters. 125 (5): 058101. arXiv:1910.00102. Bibcode:2020PhRvL.125e8101S. doi:10.1103/PhysRevLett.125.058101. ISSN 0031-9007. PMID 32794890. S2CID 203610681.

[10] Saxton, Michael J. (15 February 2007). "A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model". Biophysical Journal. 92 (4): 1178–1191. Bibcode:2007BpJ....92.1178S. doi:10.1529/biophysj.106.092619. PMC 1783867. PMID 17142285.

[11] Zhang, Zhidong; Angst, Ueli (2020-10-01). "A Dual-Permeability Approach to Study Anomalous Moisture Transport Properties of Cement-Based Materials". Transport in Porous Media. 135 (1): 59–78. Bibcode:2020TPMed.135...59Z. doi:10.1007/s11242-020-01469-y. hdl:20.500.11850/438735. ISSN 1573-1634. S2CID 221495131.

[JeonLeijnse2013-12] Jeon, Jae-Hyung; Leijnse, Natascha; Oddershede, Lene B; Metzler, Ralf (2013). "Anomalous diffusion and power-law relaxation of the time averaged mean squared displacement in worm-like micellar solutions". New Journal of Physics. 15 (4): 045011. Bibcode:2013NJPh...15d5011J. doi:10.1088/1367-2630/15/4/045011. ISSN 1367-2630.

[13] Bruno, L.; Levi, V.; Brunstein, M.; Despósito, M. A. (2009-07-17). "Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors". Physical Review E. 80 (1): 011912. Bibcode:2009PhRvE..80a1912B. doi:10.1103/PhysRevE.80.011912. hdl:11336/60415. PMID 19658734. S2CID 15216911.

[14] Peccianti, Marco; Morandotti, Roberto (2012). "Beyond ballistic". Nature Physics. 8 (12): 858–859. doi:10.1038/nphys2486. S2CID 121404743.

[15] Richardson, L. F. (1 April 1926). "Atmospheric Diffusion Shown on a Distance-Neighbour Graph". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 110 (756): 709–737. Bibcode:1926RSPSA.110..709R. doi:10.1098/rspa.1926.0043.

[16] Cushman-Roisin, Benoit (March 2014). Environmental Fluid Mechanics. New Hampshire: John Wiley & Sons. pp. 145–150. Retrieved 28 April 2017.

[17] Berkowicz, Ruwim (1984). "Spectral methods for atmospheric diffusion modeling". Boundary-Layer Meteorology. 30 (1): 201–219. Bibcode:1984BoLMe..30..201B. doi:10.1007/BF00121955. S2CID 121838208.

[MasoliverMontero2003-18] Masoliver, Jaume; Montero, Miquel; Weiss, George H. (2003). "Continuous-time random-walk model for financial distributions". Physical Review E. 67 (2): 021112. arXiv:cond-mat/0210513. Bibcode:2003PhRvE..67b1112M. doi:10.1103/PhysRevE.67.021112. ISSN 1063-651X. PMID 12636658. S2CID 2966272.

[19] Toivonen, Matti S.; Onelli, Olimpia D.; Jacucci, Gianni; Lovikka, Ville; Rojas, Orlando J.; Ikkala, Olli; Vignolini, Silvia (13 March 2018). "Anomalous-Diffusion-Assisted Brightness in White Cellulose Nanofibril Membranes". Advanced Materials. 30 (16): 1704050. Bibcode:2018AdM....3004050T. doi:10.1002/adma.201704050. PMID 29532967.

[:0-20] Metzler, Ralf; Jeon, Jae-Hyung; Cherstvy, Andrey G.; Barkai, Eli (2014). "Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking". Phys. Chem. Chem. Phys. 16 (44): 24128–24164. Bibcode:2014PCCP...1624128M. doi:10.1039/C4CP03465A. ISSN 1463-9076. PMID 25297814.

[21] Krapf, Diego; Metzler, Ralf (2019-09-01). "Strange interfacial molecular dynamics". Physics Today. 72 (9): 48–54. Bibcode:2019PhT....72i..48K. doi:10.1063/PT.3.4294. ISSN 0031-9228. S2CID 203336692.

[22] Manzo, Carlo; Garcia-Parajo, Maria F (2015-12-01). "A review of progress in single particle tracking: from methods to biophysical insights". Reports on Progress in Physics. 78 (12): 124601. Bibcode:2015RPPh...78l4601M. doi:10.1088/0034-4885/78/12/124601. ISSN 0034-4885. PMID 26511974. S2CID 25691993.

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PROFILPELAJAR.COM

Anomalous diffusion

Classes of anomalous diffusion

Models of anomalous diffusion

See also

References

External links

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