Anomalous diffusion

Mean squared displacement ${\displaystyle \langle r^{2}(\tau )\rangle }$ for different types of anomalous diffusion

Anomalous diffusion is a diffusion process with a non-linear relationship to time, in contrast to a typical diffusion process, in which the mean squared displacement (MSD), σr2, of a particle is a linear function of time. Physically, the MSD can be considered the amount of space the particle has "explored" in the system.

Unlike typical diffusion, anomalous diffusion is described by a power law,[1][2] σr2 ~ Dtα, where D is the diffusion coefficient and t is the elapsed time. In a typical diffusion process, α = 1. If α > 1, the phenomenon is called super-diffusion. Super-diffusion can be the result of active cellular transport processes. If α < 1, the particle undergoes sub-diffusion.[3]

The role of anomalous diffusion has received attention within the literature to describe many physical scenarios, most prominently within crowded systems, for example protein diffusion within cells, or diffusion through porous media. Sub-diffusion has been proposed as a measure of macromolecular crowding in the cytoplasm.

Recently, anomalous diffusion was found in several systems including ultra-cold atoms,[4] Telomeres in the nucleus of cells,[5] in single particle movements in cytoplasm,[6] and in worm-like micellar solutions.[7] Anomalous diffusion was also found in other biological systems, including heartbeat intervals and in DNA sequences.[8]

The daily fluctuations of climate variables such as temperature can be regarded as steps of a random walker or diffusion and have been found to be anomalous.[9]

In 1926, using weather balloons, Lewis Richardson demonstrated that the atmosphere exhibits super-diffusion.[10] 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 ${\displaystyle l_{m}={\kappa }z}$, where ${\displaystyle l_{m}}$ is the mixing length, ${\displaystyle {\kappa }}$ is the Von Kármán constant, and ${\displaystyle z}$ is the distance to the nearest boundary. [11] 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.[12]

Types of Anomalous diffusion

Of interest within the scientific community, when an anomalous-type diffusion process is discovered, the challenge is to understand the underlying mechanism which causes it. There are a number of frameworks which give rise to anomalous diffusion that are currently in vogue within the statistical physics community. These are long range correlations between the signals [13] continuous-time random walks (CTRW [14]) and fractional Brownian motion (fBm), diffusion of colloidal particles in bacterial suspensions,[15] and diffusion in disordered media.[16]

References

1. ^ Ben-Avraham, Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press.
2. ^ S. Havlin, D. ben-Avraham (2002). "Diffusion in disordered media". Adv. Phys. 51: 187. Bibcode:2002AdPhy..51..187H. doi:10.1080/00018730110116353.
3. ^ Ahmad Sharifi-Viand, Investigation of anomalous diffusion and multifractal dimensions in polypyrrole film, Journal of Electroanalytical Chemistry(Elsevier), 671: 51–57 (2012).
4. ^ 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:. Bibcode:2012PhRvL.108i3002S. doi:10.1103/PhysRevLett.108.093002. ISSN 0031-9007. PMID 22463630.
5. ^ 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.
6. ^ 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.
7. ^ 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.
8. ^ Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Peng, C.K.; Stanley, H.E. (1994). "Fractals in Biology and Medicine: From DNA to the Heartbeat". In Bunde, Armin; Havlin, Shlomo. Fractals in Science. Springer. pp. 49–89. ISBN 3-540-56220-6.
9. ^ Koscielny-Bunde, Eva; Bunde, Armin; Havlin, Shlomo; Roman, H. Eduardo; Goldreich, Yair; Schellnhuber, Hans-Joachim (1998). "Indication of a Universal Persistence Law Governing Atmospheric Variability". Physical Review Letters. 81 (3): 729–732. Bibcode:1998PhRvL..81..729K. doi:10.1103/PhysRevLett.81.729. ISSN 0031-9007.
10. ^ 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. doi:10.1098/rspa.1926.0043.
11. ^ Cushman-Roisin, Benoit (March 2014). Environmental Fluid Mechanics. New Hampshire: John Wiley & Sons. p. 145-150. Retrieved 28 April 2017.
12. ^ Berkowicz, Ruwim (1984). "Spectral methods for atmospheric diffusion modeling". Boundary-Layer Meteorol. 30 (1): 201-219.
13. ^ Buldyrev, S.V.; Goldberger, A.L.; Havlin, S.; Peng, C.K.; Stanley, H.E. (1994). "Fractals in Biology and Medicine: From DNA to the Heartbeat". In Bunde, Armin; Havlin, Shlomo. Fractals in Science. Springer. pp. 49–89. ISBN 3-540-56220-6.
14. ^ Masoliver, Jaume; Montero, Miquel; Weiss, George H. (2003). "Continuous-time random-walk model for financial distributions". Physical Review E. 67 (2). arXiv:. Bibcode:2003PhRvE..67b1112M. doi:10.1103/PhysRevE.67.021112. ISSN 1063-651X.
15. ^ Wu, Xiao-Lun (2000-01-01). "Particle Diffusion in a Quasi-Two-Dimensional Bacterial Bath". Physical Review Letters. 84 (13): 3017–3020. Bibcode:2000PhRvL..84.3017W. doi:10.1103/PhysRevLett.84.3017.
16. ^ S. Havlin, D. ben-Avraham (2002). "Diffusion in disordered media". Adv. Phys. 51: 187. Bibcode:2002AdPhy..51..187H. doi:10.1080/00018730110116353.