Braid dynamics

The so-called ferrofluids are liquids, which behave as if they were liquid magnets. They can be manipulated by external magnetic fields, like that of a small handheld bar magnet. In reality, the ferrofluid consists of a suspension of tiny, nanometer sized, permanently magnetic particles in an ordinary fluid like water or kerosene. When nonmagnetic bodies, like small plastic spheres of micrometer size, often called microspheres, are mixed into a ferrofluid and are acted upon by the field from a magnet, they will behave as if they were magnetic. Using an optical microscope and a video camera, we can study the behaviour of such particle suspensions.  Small groups of microspheres will form chains aligned along the direction of the magnetic field, as shown in the figure below, and behave almost as compass needles.

If we move the magnet further away from our sample and thereby reduce the strength of the magnetic field acting on the microspheres, the initially straight chains start to flex and vibrate. The width of a chain of spheres increases gradually with time as shown in the next figure. This happens because the spheres are continuously bombarded by the moving molecules of the fluid. The width of a chain will first increase relatively fast, but then slower and slower with time. Finally, it will reach a stable width, which is determined by the magnetic forces acting along the chain (similar to the tension in a stretched rubber band). If the microspheres were free to move, the chain would brake apart very soon. This process of motions in random directions due to molecular collisions is called diffusion. However, since the spheres also feel the magnetic field, their random motion is slowed down. This process may be called sub-diffusion. Similar restricted particle motions giving rise to sub-diffusion can be found in many liquids containing chain-like molecules, like polymers in solution.

If we instead of changing the strength of the magnetic field, change the direction of the field, i.e., by rotating a bar magnet around its centre axis, the aligned microspheres will try to keep aligned with the field. However, due to friction forces in the fluid, they may not manage to do that and will instead reorganize into complex structured patterns. They will perform some sort of complex dance along with the field. The snapshots in the figure below show some steps in the dance performed by seven micropsheres in a rotating magnetic field. The process may seem more or less random but if analyzed properly, one can often find that the microparticles move around faster than what one would expect for ordinary random motions. This increased diffusion-like motion is an example of super-diffusion.

Using tools from a sub-field of mathematics that is called braid mathematics we are able to describe the motions of the microspheres. By stacking all the two-dimensional (x-y) snapshots after each other, forming a three-dimensional structure with the time (in seconds) running along the z-axis (an x-y-t diagram), one can find the imaginary traces of the spheres. Their space-time diagram can then be constructed. This is illustrated in the upper part of next figure. The lower part of this figure shows how a small section of the diagram can be transformed into a two-dimensional pattern of under- and overcrossings of the strands (the particle trajectories). The strands have been labelled 1-7, and one introduces a symbol (in this case the Greek letter sigma) to denote that one string crosses over another. The subscript of the symbol tells the number of the string which crosses over the next one. In such a way the whole pattern can be transformed into a series of symbols or characters. One might also use the characters A, B, C, instead of the symbol with subscript. It is then easier to see that one gets a word, the braidword for that particular motion ("DEABAEFEABA" for the case shown here). It is of course much easier to send this "word" to somebody who needs to know the pattern than to send the whole picture with the space-time diagram. The original pattern may easily be regenerated from a sequence of such words.

By analysing the braid patterns formed by microspheres moving in ferrofluids, we can collect a lot of words. If we analyze how often the different braidwords occur, we find, as in ordinary spoken languages, that some words are used more frequently than others (like "the", "and", "of" in English). Indeed, if we rank the braidwords and give the most frequently occurring word rank R=1, the next most frequent rank R=2, etc., we find that the frequency of occurrence F of a word is inversely proportional to its rank, i.e., F = constant/R. This is similar to what the linguists, who study the structure of languages, have found for most common languages (e.g. English, Norwegian, Latin). This Zipf law was discovered by G.K. Zipf in 1949. So, do the moving microspheres "speak" a language? - In a way they do. By using the correct number of microspheres and appropriate magnetic field conditions, and waiting long enough (!), one could in principle find that the dancing microspheres could "write" readable text.