Diffusion

2. Random Walk

Basics

In the main text I made a lot of noise about "random walk". The text book paradigm for this always was the "drunken sailor", running around at random in two dimensions. In our enlightened times we must augment this by drunken cheerleaders to mainstream the genders in a politically correct way.

So imagine the drunken person of your choice to come out of the bar and to proceed in a haphazard way. To make it easier, we imagine that the drunken person makes steps of always the same length a, and moves with equal likelihood forward, backward, left, or right. Maybe there is a chess board pattern on the plaza, and the drunken person tries to stay in the squares; whatever. The level of intoxication is such that the drunken person makes a total of N steps and then lies down and goes to sleep. The figure below illustrates this.


Slightly idealized walk of drunken person.

You follow a first individual and record the distance L₁that your first specimen covered; the distance between the start position (exit of bar) and stop position (where the cheerleader beds down to sleep).
Now you watch the next guy coming out of the bar; your specimen No. 2. You probably will find this guy in a completely different distance L - closer to the bar or further away - in comparison to the first one. You will never find any of them at a distance larger than N · a for obvious reasons, though.
Now you do the same thing for No. 3, No 4, and a lot more. Then you determine the average distance covered by all those helpless people. This average distance we call the diffusion length L(N, a) of the random walk with step length a and number of steps N.

In contrast to an individual distance L_i, the average L is a well-defined quantity, just like the average income or the average number of words your wife needs to say that she feels slightly put out about you forgetting her birthday. It just can be very time consuming or unpleasant to measure those averages. A good theory would be rather helpful.

You will be delighted to hear that a good theory does exist (for diffusion lengths, not for your wife). While it is not all that easy to derive the relevant equations, the final result is extremely simple. Here it is:

L²(2-dim)	=	2 · a² · N

L²(3-dim)	=	3 · a² · N

L	»	a ·	N ^½

If we forget about the square root of 1, 2 or 3 for 1-dim., 2-dim. or 3-dimensional random walks (the latter done by drunken sea gulls or sharks), the theory says unambiguously that the diffusion length, the average distance between start and stop, is

proportional to the distance a covered in one step, and
proportional to the square root of the number of steps made.

Great - but how about if the step length of the drunken animal is not the same at every step but varies? Then take a to be the average step length.
You might come up with more questions like this but rest assured that the simple equation above covers every conceivable situation; we do not need to worry about details.

This is of immediate interest to sword smiths. Let's assume that you want some carbon to move into your blade from the outside, in order to induce some case hardening. The carbon atoms will do a random walk and, after some time t₁, will be "1 µm" deep inside the steel. Of course, some atoms are deeper than that, and some not quite that deep. The "1 µm" refers to some average, i.e. the the diffusion length of your carbon atoms for the conditions chosen.

If you want to double this diffusion length, you have to increase the holding time fourfold; a tenfold diffusion length increase involves a hundred-fold time, and so on. It is evidently not so efficient to increase the time, you should think about increasing the temperature.

Now let's do another simple experiment. You don't have to get up and get out, watching drunken sailors and cheer leaders stumbling around outside bars, you can do it right at home. All you need is a ruler and two coins.

Put one coin in the middle of the ruler, e.g. on the 8'' mark for a 16'' ruler. Flip the other coin, and move the first coin 1'' to the right for head, and 1'' to the left for tale.
Write down the distance of the first coin from the starting position after you did 100 tosses and thus hundred left or right movements.
Repeat this a hundred times or so, then calculate the average distance from the starting position for movements to one side - left or right.
This is the diffusion length in a 1-dim. random walk. It should come out as predicted by the equation above.
You can do even more. Produce a graph that shows how many coins ended up 1'' to one side, 2'' to one side and so on. That would be the distribution function of the distances covered.

You must admit that this is far easier to do than to follow drunken cheer leaders. Maybe more boring, but easier.
Of course, if you know a bit about computer programming, you can force your PC or notebook to do the work for you. Let the computer flip a digital coin and move a point on the screen one step to the left or right, accordingly. Actually, the computer won't mind to do that for a lot of points simultaneously and follow every one of them with total precision, so you can get good statistics rather quickly

If you are not so sure about writing computer programs, you can take the easy way out and use the program that two of my students (Niclas Köser und Sören Witt) have written for you. Here it is:

Random Walk Simulator

The program starts with a bunch of red points, all of which sit at the zero point of the "ruler". Ignore everything except the "Run", "Stop" and "Reset" buttons, and just start playing with it.

The great drunken red-dot race

Pressing the "Run" bottom induces the computer to flip a digital coin for every point and move it left or right. You can see that nicely by looking at the front runners.

Above the cloud of moving dots the graph mentioned above is constructed. Obviously, with time a Gaussian Bell curve develops - what else could it be?

The program can also be used to simulate the (1-dim.) movement of real atoms that jump over some energy barrier. That's what all those boxes containing adjustable numbers are there for. Ignore them.

6.3 Random Walk