38. A random walk in the plane

39. A Brownian path in the plane

Brownian motion is named after the Scottish botanist Robert Brown (1773–1858) who in 1827 observed that minute particles of pollen in water moved on highly irregular paths. This was eventually explained as the result of the particles being continually bombarded in a random manner by molecules of water. Except at the finest scales, the paths followed in this ‘real’ Brownian motion are very similar indeed to those seen in the random walk construction. The effect of the particle being struck by molecules travelling in essentially random directions corresponds to the random choice of direction in the random walk. In 1905, Albert Einstein (1879–1955) showed that Brownian motion could be described by an equation of a similar form to the equation describing the diffusion of heat. This involved a term that could be related to measurable physical quantities and led to the determination of the size of atoms and the weight of molecules of various gases. Einstein’s predictions were verified experimentally in 1913 by Jean Perrin (1870–1942) who won a Nobel Prize for this work. However, it was not until 1923 that Norbert Wiener (1894–1964) put Brownian motion on a completely rigorous mathematical basis.

Brownian motion is a very good example of a natural fractal phenomenon that can be explained very well by a simple mathematical model. Even so, the description inevitably breaks down at a very fine scale—a mathematical Brownian path continually changes direction to such an extent that it would require an infinite amount of energy for a particle of positive mass to follow such a path exactly.

Fractal fingering

Random walks and Brownian motion underlie a mechanism for fractal growth known as fractal fingering or, more formally, diffusion limited aggregation, which was proposed by physicists T. A. Witten and L. M. Sander in 1981. Fractal fingering occurs in a range of physical settings where there is some form of random diffusion, from forked lightning to bacterial growth on food.

Take a large disc in the plane, and site a small ‘seed’ in the centre. A small particle is released from a random point on the perimeter of the disc, and follows a random walk or Brownian path inside the disc until it touches the seed, whence it sticks to the seed. Another particle is released and moves randomly until it hits and sticks to the existing deposit. Further particles follow random paths and hit and stick to the existing cluster. Continuing with many thousands of small particles, the deposit grows outwards from the initial seed. Figure 40 indicates schematically how this process is simulated on a computer—the particles are squares on a very fine grid that follow a random walk on the grid, with equal probability of moving north, south, east, or west, until they stick at a grid square next to one that is already occupied. Although the growth is random, it assumes a characteristic appearance of repeatedly branching strands fingering outwards, Figure 41. These ‘fractal fingers’ or ‘dendrites’ tend to grow from near their ends, since when a new particle comes in on a random path it is much more likely to hit the cluster near its outer extremities than to weave its way in between the fingers to be depos how to move around the planeNixited nearer the centre. The fine structure and fractal nature of such a growth is apparent, and box-counting estimates of the dimension are always close to 1.70, or about 2.43 for the analogous growth process in space.

Crystal growth by electrolysis is a good practical example of fractal fingering. A circular dish is filled with a shallow layer of copper sulphate (CuSO₄) solution, a copper wire is suspended in the centre of the dish, and a copper strip curved into a circle around the perimeter. If a battery of a few volts is connected with the negative to the centre wire and the positive to the perimeter wire, then a deposit of copper grows outwards from the centre and after an hour or so has a branched form very similar to Figure 41 perhaps with slightly thicker fingers. The chemical process taking place is very similar to that described above. In solution copper sulphate molecules split into copper Cu²⁺ ions and sulphate ions which move around following random Brownian-like paths. When the voltage is applied the copper ions that hit the cathode receive two electrons and are deposited as copper. The copper deposit grows outwards with copper being deposited as the randomly moving copper ions hit the existing growth. Electrolysis of zinc sulphate (ZnSO₄) using a carbon electrode in the centre and zinc around the perimeter of the dish behaves in a like manner, with zinc fingers growing out from the centre.

40. Computer simulation of fractal fingering: successive particles are released from random points A on the circle and perform random walks until they reach a square next to one already shaded, B. This square is shaded and the process repeated, so the shaded squares grow out from the initial ‘seed’ square S