Anomalous diffusion explained

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 =2 dD\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} _{\alpha\tau} 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.

See also

• • • • Long term correlations

References

• 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 . 10.1529/biophysj.104.044263. 15339818 . 2004BpJ....87.3518W . 1304817 .
• Bouchaud . Jean-Philippe . Georges . Antoine . 1990 . Anomalous diffusion in disordered media . 10.1016/0370-1573(90)90099-N . Physics Reports . 195 . 4–5. 127–293 . 1990PhR...195..127B .
• von Kameke . A.. 23202701. et al . 2010 . Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow . Phys. Rev. E . 81 . 6 . 066211. 10.1103/physreve.81.066211 . 20866505. 2010PhRvE..81f6211V .
• 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 . 10.1016/j.camwa.2009.08.020. free .
• 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 . 10.1016/j.advwatres.2012.11.005. 23794783 . 2013AdWR...52..292S . 3686513 .
• 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.. en. 16. 44. 24128–24164. 10.1039/c4cp03465a. 25297814. 1463-9076. 2014PCCP...1624128M. free.

External links

• Boltzmann's transformation, Parabolic law (animation)
• Anomalous interface shift kinetics (Computer simulations and Experiments)

Notes and References

1. Einstein. A.. 1905. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik. de. 322. 8. 549–560. 10.1002/andp.19053220806. free.
2. von Smoluchowski. M.. 1906. Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Annalen der Physik. de. 326. 14. 756–780. 10.1002/andp.19063261405.
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 . 1109.1503 . 2012PhRvL.108i3002S . 10.1103/PhysRevLett.108.093002 . 0031-9007 . 22463630 . 24674876.
4. Saporta-Katz . Ori . Efrati . Efi . 2019 . Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System . Physical Review Letters . 122 . 2 . 024102 . 1706.09868 . 10.1103/PhysRevLett.122.024102 . 30720293 . 119240381.
5. Colbrook . Matthew J. . Ma . Xiangcheng . Hopkins . Philip F. . Squire . Jonathan . 2017 . Scaling laws of passive-scalar diffusion in the interstellar medium . . 467 . 2 . 2421–2429 . 1610.06590 . 2017MNRAS.467.2421C . 10.1093/mnras/stx261 . 20203131.
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 . 2009PhRvL.103a8102B . 10.1103/PhysRevLett.103.018102 . 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 . en . 108 . 16 . 6438–6443 . 2011PNAS..108.6438W . 10.1073/pnas.1016325108 . 0027-8424 . 3081000 . 21464280 . free.
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 . 2013BpJ...104.1652R . 10.1016/j.bpj.2013.01.049 . 0006-3495 . 3627875 . 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 . en . 125 . 5 . 058101 . 1910.00102 . 10.1103/PhysRevLett.125.058101 . 0031-9007 . 32794890 . 203610681.
10. Saxton . Michael J. . 15 February 2007 . A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model . Biophysical Journal . 92 . 4 . 1178–1191 . 2007BpJ....92.1178S . 10.1529/biophysj.106.092619 . 1783867 . 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 . en . 135 . 1 . 59–78 . 10.1007/s11242-020-01469-y . 1573-1634 . 221495131 . free. 20.500.11850/438735 . free .
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 . 2013NJPh...15d5011J . 10.1088/1367-2630/15/4/045011 . 1367-2630 . free.
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 . 10.1103/PhysRevE.80.011912. 19658734 . 11336/60415 . 15216911 . free .
14. Peccianti . Marco . Morandotti . Roberto . Roberto Morandotti . 2012 . Beyond ballistic . Nature Physics . 8 . 12 . 858–859 . 10.1038/nphys2486. 121404743 .
15. Richardson. L. F.. Atmospheric Diffusion Shown on a Distance-Neighbour Graph. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 1 April 1926. 110. 756. 709–737. 10.1098/rspa.1926.0043. 1926RSPSA.110..709R . free.
16. Book: Cushman-Roisin. Benoit. Environmental Fluid Mechanics. March 2014. John Wiley & Sons. New Hampshire. 145–150. 28 April 2017.
17. Berkowicz. Ruwim. Spectral methods for atmospheric diffusion modeling. Boundary-Layer Meteorology. 1984. 30. 1. 201–219. 10.1007/BF00121955. 1984BoLMe..30..201B. 121838208.
18. Masoliver . Jaume . Montero . Miquel . Weiss . George H. . 2003 . Continuous-time random-walk model for financial distributions . Physical Review E . 67 . 2 . 021112 . cond-mat/0210513 . 2003PhRvE..67b1112M . 10.1103/PhysRevE.67.021112 . 1063-651X . 12636658 . 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. 10.1002/adma.201704050 . 29532967. free .
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. . en . 16 . 44 . 24128–24164 . 2014PCCP...1624128M . 10.1039/C4CP03465A . 1463-9076 . 25297814 . free.
21. Krapf. Diego. Metzler. Ralf. 2019-09-01. Strange interfacial molecular dynamics. Physics Today. en. 72. 9. 48–54. 10.1063/PT.3.4294. 203336692 . 0031-9228.
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. 10.1088/0034-4885/78/12/124601. 26511974. 25691993 . 0034-4885.