Anomalous diffusion

Anomalous diffusion is in statistical physics, a particular type of transport process of diffusion or Brownian molecular motion that occurs in many complex (eg, viscoelastic ) fluids. You can not be described by the ordinary ( Fickian ) diffusion law. In contrast to normal diffusion, the mean square displacement of a diffusing particle abnormal, that is the space which the particle travels through in the time to grow, not proportional to, but typically follows a power law of anomaly parameters α. Anomalous diffusion describes random walks with long - reichweitigen correlations for which no longer applies the central limit theorem of statistics. Such transport processes occur, for example, in cells or at the travel behavior of people.

3.1 Normal diffusion
3.2 Lévy flights
3.3 Continuous time random walks ( CTRW )

4.1 anomalous diffusion and the Langevin equation
4.2 Fractional diffusion equation

Definition and properties

Definition

Usual diffusion processes can be macroscopically described by the Fick's diffusion equation. Microscopically does this description into a Brownian motion ( Wiener process ) over, with the mean square displacement:

The factor n is the number of spatial dimensions and the parameter D is the diffusion coefficient.

Anomalous diffusion, distinguishes itself by the following dependence:

Here Ka is a generalized diffusion coefficient and α the anomaly parameter. The unit of this generalized diffusion coefficient is [ Ka ] = m2/sα, so depends on the anomaly parameter. There are two regimes, which are also shown in the figure at the top:

Subdiffusion (): This type of slow diffusive motion can be observed around inside cells and random walks on fractal structures.
The special case describes the ordinary normal diffusion.
Superdiffusion (): This accelerated diffusion occurs eg in Lévy flights on, or about the movement of bills or travel movement of people.
The special case is called ballistic diffusion (English ballistic diffusion ). This corresponds to a case where in addition to the diffusion movement is also present a drift.

Anomalous diffusion as a macroscopic effect

Anomalous diffusion itself is initially a macroscopic effect. As shown by the various examples above already, deriving the microscopic cause of the anomalous diffusion is not easy.

Time-dependent diffusion coefficient and memory

The mean square displacement can be expressed formally by a time-dependent diffusion coefficient d.alpha ( τ ):

The diffusion coefficient is no longer constant over time, the behavior (the " diffusion rate " ) of a particle thus depends on how long it's been moved ( for subdiffusion example it is slower and slower the longer it moves ). This means that almost a memory exists in the system, which makes the actual motion of the history -dependent. A more detailed mathematical model of this will be further described below in the section, and the anomalous diffusion Langevin equation.

Occurrence of anomalous diffusion

Anomalous diffusion phenomena occur in different systems. Some examples are to be combined, which will be explained partially in the rest of Article closer:

Superdiffusion with α > 1: in the theoretical random-walk model of the Lévy- flight
In the movement of notes or the travel movement of people. Travelers people remain, for example, typically some time in a city and move there on a small spatial scale. With a certain ( low but non-zero ) probability they do then a trip to a distant city, resulting in a big jump. Lévy flights are a theoretical model for such behavior.
Movement of individual cells in cell aggregates

Subdiffusion with 0 < α <1: Inside cells is observed subdiffusion during movement of macromolecules through the cytoplasm. One reason for this may be the so-called molecular crowding, so the presence of many ( close-packed ) macromolecules and organelles in the cytoplasm
Anomalous diffusion is also observed on the membranes of cells. The cell membrane is here a complex system made up of many different components ( see, eg, fluid mosaic model).
Random walks on fractal structures such as Perkolationsklustern. This can be demonstrated experimentally by NMR Diffusivitätsmessungen in porous systems.
Diffusion in polymer networks
Monomerbewegung the long polymers such as DNA, is limited to the characteristic time scale also abnormal diffusion, here triggered by a limited internal movement of the polymer ( see, for example, the simple model of the Rouse polymer dynamics).
Charge carrier transport in amorphous semiconductors

Theoretical description by Random Walks

Normal diffusion

As already mentioned, show certain random walk processes an anomalous diffusive behavior. This one describes the progression of (in this example one-dimensional ) motion in discrete time steps? T. The position jump from one time step to the next is Gaussian distributed for normal diffusion:

This characteristic Gaussian distribution is considered due to the central limit theorem of statistics for many operations. However, there are, as in the following examples, its conditions are no longer met (eg if the variance of the above distribution can not be defined ), one can observe anomalous diffusive behavior.

Lévy flights

Anomalous superdiffusion occurs in random walk processes where the jump length distribution is endlastig. Here the central limit theorem does not hold, since the variance of endlastigen distributions diverges. One example is the already mentioned Lévy flights where rare ( but more often than in a Gaussian distribution ) can occur very long jumps. The jump length distribution takes here with a power law:

In the picture a few steps of such a process are shown. The rare large jumps are easy to recognize.

Continuous time random walks ( CTRW )

Another random walk process with anomalous diffusive characteristics are so-called continuous time random walks ( CTRW ). The movement is not divided into equal time steps? T, but with a constant jump length Ax the waiting time between two jumps from a distribution is considered. One can interpret as a diffusion on a lattice with traps that too, where the traps can capture the diffusing particles of different lengths. If the waiting time distribution endlastig, so:

So this also leads to anomalous subdiffusion with anomaly parameter α.

Continuous theoretical models

Anomalous diffusion and the Langevin equation

Normal diffusing particles in a viscous medium can be described by the Langevin equation:

Where x (t) of particle location at time t, the friction coefficient and ξ Fst a statistical correlation with vanishing force, so white noise. This stochastic differential equation can be generalized to the fractional Langevin equation:

Where K ( τ ) is a so-called memory kernel is now ( German about memory convolution kernel ) induced one ( also long-range ) temporal coupling. The motion of the particle therefore also depends on its past ( integral) from that at normal Brownian motion was not the case (which corresponds to a non- Markov random walk ). Referring now in particular again a power law for K ( τ ) to so

As also follows from this approach, an anomalous mean square displacement with anomaly α. This approach can be used to model anomalous diffusion, as occurs in viscoelastic media.

Fractional diffusion equation

With the help of the defined in the Fractional Calculus fractional integro-differential operators can often relied on modeling normal diffusion phenomena Fokker -Planck equation for anomalous diffusion expand. This ( then fractional ) differential equation describes the time evolution of the probability density W ( x, t) of diffusing particles x at time t and location.

Here, the Riemann Liouville operator clearly than the α -th derivative of the function f (t ) after the time defined by the integral representation:

Here, Γ ( x) is the gamma function. The solution of this fractional differential equation again leads to the anomalous mean square displacement: