Diffusion Part 2: Population Dynamics

  I wouldn’t blame you for wondering how we can say anything predictive at all about systems of particles that move randomly. It’s important to recognize that while the motion of an individual particle is random and unpredictable, the macroscopic behaviour of a system as a whole is still very much predictable. Say for example that you flip a thousand coins. While you can’t predict the outcome of any one coin flip, you can still be quite certain that you’ll get roughly five hundred heads and five hundred tails. While this is not the only possible outcome, it is certainly the most likely one; quantifying the likelihood of deviating from this outcome is one of many uses for what is called the Normal (aka Gaussian) distribution. Normal distributions describe data sets that are randomly and symmetrically distributed about some average value. Some examples include heights of people, shoe sizes, errors in measurements, the grades on an exam (the famous bell curve), and of course the positions of drunken walkers. Click below to start a simulation of 1000 drunken walkers.

Pay particular attention to the plot in the bottom left corner: it’s a histogram that tracks the number of particles N as a function of position x. Notice that the distribution of particle positions forms a bell-shaped curve that broadens with time. This is a normal distribution!

One way to quantify the “speed” of the diffusion is to measure how the root-mean-squared displacement Rrms of the population changes with time. While the mean displacement of the population is zero (most particles hover around x=0), Rrms is non-zero. It is a measure of the width of the normal distribution. More specifically, it’s the distance that encompasses 68% of the area of the curve, or in other words the distance that a particle has a 32% chance of exceeding in some time interval. In practice, we can use Rrms as a measure of how far roughly two thirds of a population has diffused in a given time.

In the simulation above, Rrms is computed two different ways. In red, we painstakingly calculate Rrms by summing the squared displacements of all the particles in the system, dividing by the number of particles, then square-rooting the result. In blue, we simply take the square root of twice the number of timesteps that have elapsed. You can see that the two methods agree! Stay tuned for a proof of this result. The take-home message is that Rrms increases like the square root of time, so a random walk is not a particularly efficient way to explore space. Nevertheless, cells rely on this mechanism for short-scale passive transport as it does not require any external energy input.

In part 3, we take a look at how complex (and beautiful) structures can emerge in systems of drunken particles.