# Anomalous Diffusion

This is part of my “journal club for credit” series. You can see the other computational neuroscience papers in this post.

## Unit: Diffusion

Organized by Ben Regner

1. Standard Diffusion
2. Anomalous Diffusion
3. Life at Low Reynold’s Number

## What is anomalous diffusion?

If one measures the mean square displacement vs time, it can be parameterized as

$< x^2> = t^\alpha$

where $\alpha=1$ is Brownian (standard diffusion), $0<\alpha<1$ is subdiffusive, $1<\alpha<2$ is superdiffusive, and ballistic is $\alpha=2$. So the technical definition of anomalous diffusion is $0<\alpha<1$ or $1<\alpha<2$.

## How to describe anomalous diffusion?

Currently, there is no “best” or “simple” description of anomalous diffusion in the general case. However, continuous-time random walks (CTRW) are one paradigm that I find helpful as a conceptual and simulation framework.

In the simplest discrete random walk (DRW), at every time step, a particle makes a jump of fixed size, the only question is the direction. The next generalization has the particle make a jump at every time step, but now it draws the jump size from a distribution.

The idea of a CTRW is that there is now a distribution both of the waiting time between jumps, and the jump size. If the waiting time follows the exponential distribution and the jump size follows the normal distribution, one ends up with the Wiener process aka standard diffusion and Brownian motion.

## What causes anomalous diffusion?

Just as a reminder, there are three conditions that need to be satisfied for Brownian motion (standard diffusion):
1. Increments are independent
2. Increments are wide sense stationary. 1st moment and autocovariance don’t depend on time (this is weaker condition then complete stationarity)
3. Zero mean

The third condition is often ignored by examining the motion relative to the mean displacement (ie the actual displacement is not Brownian, but fluctuations in the displacement could be Brownian). So really, the first two are the more important conditions. Therefore, anomalous diffusion arises due to non-independent increments and/or correlations in time of the mean and/or standard deviation.

The CTRW allows one to think more precisely about different mechanisms that can give rise to anomalous diffusion. There is not one single way to get sub or super-diffusion in CTRW, since there are two, potentially dependent, distributions (waiting time and jump size). However, there are a few common situations that seem to arise often in biology and elsewhere (see Random walk models in biology, Box 2 for original idea). Subdiffusion in biology is often caused by longer waiting time distributions (compared to exponential), or molecular crowding, while superdiffusion in occurs when jump sizes are drawn from a Levy flight or other alpha stable distributions.

## Examples

For further exploration of anomalous diffusion in biology, I recommend these papers

• This is an interesting paper that introduces a renormalization group approach to classifying diffusion processes

# Standard Diffusion

This is part of my “journal club for credit” series. You can see the other computational neuroscience papers in this post.

## Unit: Diffusion

Organized by Ben Regner

1. Standard Diffusion
2. Anomalous Diffusion
3. Life at Low Reynold’s Number

## Papers

Brownian Motion. By Einstein in 1905.

Brownian Motion. By Langevin in 1908.

An Introduction to Fractional Diffusion. By Henry, Langlands, and Straka in 2010.

## What is diffusion?

Diffusion is the general process by which small particles move from regions of high concentration to low concentration. Check out the link to the Wikipedia articles above for some cool videos and animations. Diffusion is extremely ubiquitous and plays an essential role in biology. For example, oxygen diffuses from your lungs to unoxygenated blood, which then delivers it to the rest of your body where it diffuses out of your blood and into your cells. Additionally, signals between neurons are transmitted by several different diffusing molecules.

Mathematically, standard diffusion is described by two fundamental equations.

Fick’s First Law: Particles move from high-to-low concentration.

$j=-D\frac{\partial n}{\partial x}$

where $n$ is the number of particles, $x$ is the location of the particles, $D$ is the diffusion constant, and  $j$ is the flux of particles.

Fick’s Second Law: Conservation of particles combined with Fick’s First Law leads to the diffusion equation.

If particles cannot be created or destroyed, they follow a conservation law:

$\frac{\partial n}{\partial t} = -\frac{\partial j}{\partial x}$

Combining the conservation law with Fick’s First Law gives us the diffusion equation:

$\frac{\partial n}{\partial t} = D \frac{\partial^2 n}{\partial x^2}$

## Brownian Motion

In 1827 Robert Brown looked at pollen in water under a microscope, see Wikipedia page for simulations of the observations. Much to his surprise, the pollen acts as if it alive! Brown verified that pollen is not alive and any small, inorganic particle followed similar motion. In 1905, during Einstein’s miracle year, he wrote a paper on an atomistic description that describes Brownian Motion. In 1908 Langevin used a different approach (that is “infinitely simpler” in his words) to describe Brownian motion. The general explanations are outlined below.

1. Einstein’s Derivation

Einstein’s goal was a probability based description of Brownian motion that connects to Fick’s law. Einstein makes several assumptions about the particles, including

In the end, Einstein finds a solution that is Gaussian, implying that the mean square displacement is linear in time for Brownian motion:

$< x^2> = t$

More generally, the mean square displacement could depend on some power of time, usually parameterized as

$< x^2> = t^\alpha$

where $\alpha=1$ is Brownian, $0<\alpha<1$ is subdiffusive, $1<\alpha<2$ is superdiffusive, and ballistic is $\alpha=2$. Note, one can get up to $\alpha=3$ in certain turbulent regimes.

2. Langevin’s Derivation
The Langevin approach is to start with a particle based description. The first assumption is the equipartition theorem to determine the kinetic energy (KE)
$KE = \frac{k_B T}{2} = m (\frac{d^2 x}{dt^2})^2$

Then, one looks at the actual forces on the particle:

KE = Stoke’s + stochastic variable
$m (\frac{d^2 x}{dt^2})^2 = -6 \pi \eta r \frac{dx}{dt} + X$
where $X$ is a stochastic variable. It is assumed to be zero mean, unit variance, and no time correlations, aka white noise.

After multiplying both sides of the equation by x, doing some algebra, and then taking the average solution, one arrives at the same results as Einstein (after ignore a short time transient).

3. Random Walk Derivation.

There is a third way to derive Brownian motion that is layed out in the book chapter above. The idea is to look at a single particle and do a microscopic random walk. One can set up a recursive definition that defines a binomial probability solution. After a large number of steps, the central limit theorem applies and we end up with a Gaussian solution.

How do we get Brownian motion?

In general, there are three conditions that need to be satisfied for Brownian motion:
1. Increments are independent
2. Increments are wide sense stationary. 1st moment and autocovariance don’t depend on time (this is weaker condition then complete stationarity)
3. Zero mean

The third condition is often ignored by examining the motion relative to the mean displacement (ie the actual displacement is not Brownian, but fluctuations in the displacement could be Brownian). So really, the first two are the more important conditions.

## Fundamental Questions

• Einstein made three major assumptions in his derivation. 2/3 are often violated by biology, which assumption is relatively safe?
• What biological processes do you think are actually diffusive vs sub/super-diffusive? Think about the 3 conditions for Brownian motion listed above. Note, this is a preview for the next post.