Diffusion process with a non-linear relationship to time
Mean squared displacement
⟨
r
2
(
τ
)
⟩
{\displaystyle \langle r^{2}(\tau )\rangle }
for different types of anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD),
⟨
r
2
(
τ
)
⟩
{\displaystyle \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,
⟨
r
2
(
τ
)
⟩
=
2
d
D
τ
{\displaystyle \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]
^ 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. 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. 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 . 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 . {{cite journal }}
: CS1 maint: unflagged free DOI (link )
^ 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 . 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. 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 .