Anomalous diffusion

Unlike typical diffusion, anomalous diffusion is described by a power law,

$$\backslash langle\; r^\{2\}(\backslash tau\; )\backslash rangle\; =K\_\backslash alpha\backslash tau^\backslash alpha\backslash ,$$

where $K\_\backslash alpha$ is the so-called generalized diffusion coefficient and $\backslash tau$ 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.
• : superdiffusion. Superdiffusion can be the result of active cellular transport processes or due to jumps with a heavy-tail distribution.{{Cite journal |last1=Bruno |first1=L. |last2=Levi |first2=V. |last3=Brunstein |first3=M. |last4=Despósito |first4=M. A. |date=2009-07-17 |title=Transition to superdiffusive behavior in intracellular actin-based transport mediated by molecular motors |url=https://link.aps.org/doi/10.1103/PhysRevE.80.011912 |journal=Physical Review E |volume=80 |issue=1 |pages=011912 |doi=10.1103/PhysRevE.80.011912|pmid=19658734 |bibcode=2009PhRvE..80a1912B |hdl=11336/60415 |s2cid=15216911 |hdl-access=free }}
• α = 2: ballistic motion. The prototypical example is a particle moving at constant velocity: $r\; =\; v\backslash tau$.
• $\backslash alpha\; >\; 2$: hyperballistic. It has been observed in optical systems.{{cite journal |last1=Peccianti |first1=Marco |last2=Morandotti |first2=Roberto |author-link2=Roberto Morandotti |year=2012 |title=Beyond ballistic |journal=Nature Physics |volume=8 |issue=12 |pages=858–859 |doi=10.1038/nphys2486|s2cid=121404743 }}

In 1926, using weather balloons, Lewis Fry Richardson demonstrated that the atmosphere exhibits super-diffusion.{{cite journal|last1=Richardson|first1=L. F.|title=Atmospheric Diffusion Shown on a Distance-Neighbour Graph|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|date=1 April 1926|volume=110|issue=756|pages=709–737|doi=10.1098/rspa.1926.0043|bibcode = 1926RSPSA.110..709R |doi-access=free}} 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 $l\_m=\{\backslash kappa\}z$, where $l\_m$ is the mixing length, $\{\backslash kappa\}$ is the von Kármán constant, and $z$ is the distance to the nearest boundary.{{cite book|last1=Cushman-Roisin|first1=Benoit|title=Environmental Fluid Mechanics|date=March 2014|publisher=John Wiley & Sons|location=New Hampshire|pages=145–150|url=https://engineering.dartmouth.edu/~d30345d/books/EFM.html|access-date=28 April 2017}} 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.{{cite journal|last1=Berkowicz|first1=Ruwim|title=Spectral methods for atmospheric diffusion modeling|journal=Boundary-Layer Meteorology|date=1984|volume=30|issue=1|pages=201–219|doi=10.1007/BF00121955|bibcode=1984BoLMe..30..201B|s2cid=121838208}}

The types of anomalous diffusion given above allows one to measure the type. 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){{cite journal |last1=Masoliver |first1=Jaume |last2=Montero |first2=Miquel |last3=Weiss |first3=George H. |year=2003 |title=Continuous-time random-walk model for financial distributions |journal=Physical Review E |volume=67 |issue=2 |pages=021112 |arxiv=cond-mat/0210513 |bibcode=2003PhRvE..67b1112M |doi=10.1103/PhysRevE.67.021112 |issn=1063-651X |pmid=12636658 |s2cid=2966272}} and fractional Brownian motion (fBm), and diffusion in disordered media.{{cite journal |last1=Toivonen |first1=Matti S.|last2=Onelli |first2=Olimpia D. |last3=Jacucci |first3= Gianni |last4=Lovikka|first4=Ville |last5=Rojas |first5=Orlando J. |last6=Ikkala |first6=Olli|last7=Vignolini |first7=Silvia |date=13 March 2018 |title=Anomalous-Diffusion-Assisted Brightness in White Cellulose Nanofibril Membranes

|journal=Advanced Materials |volume= 30|issue= 16|pages= 1704050|doi=10.1002/adma.201704050 |pmid=29532967|url=https://www.repository.cam.ac.uk/handle/1810/282817|doi-access=free |bibcode=2018AdM....3004050T }} 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.{{Cite journal |last1=Metzler |first1=Ralf |last2=Jeon |first2=Jae-Hyung |last3=Cherstvy |first3=Andrey G. |last4=Barkai |first4=Eli |date=2014 |title=Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking |journal=Phys. Chem. Chem. Phys. |language=en |volume=16 |issue=44 |pages=24128–24164 |bibcode=2014PCCP...1624128M |doi=10.1039/C4CP03465A |issn=1463-9076 |pmid=25297814 |doi-access=free}}{{Cite journal|last1=Krapf|first1=Diego|last2=Metzler|first2=Ralf|date=2019-09-01|title=Strange interfacial molecular dynamics|url=http://physicstoday.scitation.org/doi/10.1063/PT.3.4294|journal=Physics Today|language=en|volume=72|issue=9|pages=48–54|doi=10.1063/PT.3.4294|bibcode=2019PhT....72i..48K |s2cid=203336692 |issn=0031-9228|url-access=subscription}}{{Cite journal|last1=Manzo|first1=Carlo|last2=Garcia-Parajo|first2=Maria F|date=2015-12-01|title=A review of progress in single particle tracking: from methods to biophysical insights|url=https://iopscience.iop.org/article/10.1088/0034-4885/78/12/124601|journal=Reports on Progress in Physics|volume=78|issue=12|pages=124601|doi=10.1088/0034-4885/78/12/124601|pmid=26511974|bibcode=2015RPPh...78l4601M |s2cid=25691993 |issn=0034-4885|url-access=subscription}} This type of motion require novel formalisms for the underlying statistical physics because approaches using microcanonical ensemble and Wiener–Khinchin theorem break down.

• {{annotated link|Lévy flight}}
• {{annotated link|Random walk}}
• {{annotated link|Percolation}}
• Long term correlations{{clarify|date=December 2018}}
• {{annotated link|Long-range dependence}}
• {{annotated link|Hurst exponent}}
• {{annotated link|Detrended fluctuation analysis|abbreviation=DFA}}
• {{annotated link|Fractal}}

{{reflist}}

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• [http://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/node4.html Boltzmann's transformation, Parabolic law (animation)]
• [http://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/node13.html Anomalous interface shift kinetics (Computer simulations and Experiments)]

Category:Physical chemistry