Mark Ward

BEYOND CHAOS

The Underlying Theory Behind Life, the Universe, and Everything

Thomas Dunne Books 2001

pg 76

Keywords: Universality - fractals - degree of organisation - systems: classified by the number of dimensions they possessed -

"self-similar" systems - systems must be open - energy or information must be able to flow across their boundaries - millions of interacting units - "self-similar" systems - self-similar structures - fractals - Sierpinski carpets, Koch snowflakes, Menger sponges and Cantor dusts - seemed to be between dimensions -

power laws (The defining mark of both fractals and systems that display 1/f noise is the power law that expresses the relationship between events or structures of different magnitudes) -

dynamics of self-similar systems (systems made up of thousands, if not millions, of interacting units managed to maintain themselves along the critical boundary) - interaction drives a system to self-organise - self-organised criticality - critical point - self-organised around its critical point - origin of all the systems where 1/f noise and fractals were apparent (at the critical point the properties of the individual elements cease to matter and the interactions take over. Order emerges. Life begins.)

Universality

Vastly different systems can, in some circumstances, exhibit exactly the same degree of organisation. When systems were classified by the number of dimensions they possessed, which defines how order can spread through them, it became obvious that there were only a few categories or classes that separated them.

In planar systems, such as a bar magnet, the order can really only spread in two dimensions - along the length of the bar or across it. The behaviour of all the systems within one class or category was mathematically identical - hence the relevance of the term universal. Any theory that could be used to explain what was happening to one system in one class could apply with the same force to all the others in that class.

Considered in this way boiling water and hot magnets are indistinguishable. Sand piles with avalanches of grains running down their sides are the same as other three-dimensional systems when they are at their critical point. We can create a short list of the properties ....these systems should possess to be considered as places where Universality might crop up.

1 The systems must be open and energy or information must be able to flow across their boundaries.

2 They must be made up of thousands, if not millions, of interacting units. Only when there are several scales available to work across does Universality become easy to spot.

3 The units should have the same sort of properties as each other. Between systems these units can be very different. Water droplets in a cloud have little in common with Wall Street stock traders, but both systems are good candidates for places where Universality is at work.

4 There must be a constant source of energy or information flowing into and out of the system driving it to a state of constantly shifting change. This energy flow "tunes" the development of the system and gives its something to react to. In some cases of this flow may be the output from another system displaying Universality.

Given these characteristics is easy to see why the field became known as Universality. Once you know where to look and what to look for you can spot this stuff everywhere. The world is riddled with it.

With Universality physics has begun to tackle the real world. Too often physics is to aloof and unable to tackle the problems that matter. The things we are profoundly interested in - stock market movements, cycles of history, earthquakes, whatever - remain closed off.

However, with Universality, the desert of our ignorance begins to bloom, and with it our understanding of the world we live in and ourselves. This is the physics of edges, interfaces, organisms and humans.

pg 78

Fractals

The important idea to take on board is that at a critical point the same degree of order is present at all scales, no matter how close or far away you get. The systems showing this similarity across scales are said to possess a critical exponent. This expresses the degree of order apparent at any and all of the scales. In the late 1970s, thanks to the efforts of Benoit Mandelbrot, a new name started to be used for systems that appeared to be the same at all scales. Such "self-similar" systems started to be called fractals.

Mandelbrot realised that a myriad of real-world phenomena can be described as fractals: Phenomena that, when examined at a variety of scales, display the same degree of order. They are self-similar.

Self-similarity means that there is not one scale that can be used to get a definitive answer about the length, size or distribution of these phenomena. They look the same at all scales.

When fractals first appeared in formal mathematics they were viewed with horror because they seemed to break all the rules about dimensions. Geometry is all about structures that exist in one, two, three or more dimensions. It works with points, lines, planes and solids, but the first fractals - Sierpinski carpets, Koch snowflakes, Menger sponges and Cantor dusts - seemed to be between dimensions. At first mathematicians had no idea what to do with them.

Mandelbrot realised that much that was interesting in the world existed between dimensions. A coastline is neither a plane nor a solid structure, it exists somewhere between the two. He came up with the word fractal because he wanted to term that could do justice to the ragged shapes and structures he was studying. Nothing in formal geometry, which dealt with constructed of exactly one, two or three dimensions, could be used to describe a something as messy as a coastline or a cloud. He needed a word that could do justice to structures that had to be considered at all scales not just one.

At the critical point all systems exhibit the same degree of order at all scales. There is no one scale that matters more than the others - the all are equally important. Ferromagnets at 770°C are self-similar. So are fractals. Whenever you see a fractal you are seeing a system at its critical point.

Fractals or self-similar structures can be found everywhere in the real-world: the curling shell of the Nautilus, the overlapping plates of a pineapple skin, the coastline of Britain, mountain ranges, and even the scattering of galaxies, clusters and superclusters across the universe. There are fractals at work in music, the formation of fern leaves, rainfall patterns, fiords, DNA and the branching blood vessels in your body. Some have even suggested that turbulence is fractal and that lungs are simply turbulent patterns cast, not in stone, but in cells.

Fractals are distinguished by the fact that they have no characteristic length scale. There are also fractals that have no characteristic time scale, i.e. the pattern of events look self-similar on all time scales, whether you're studying a second, a minute or an hour. The these fractals go by the name of 1/f (pronounced one-over-f) noise. They too are popping up everywhere and have been found in huge variety of diverse systems, such as the flow of the Nile River, the pulses of light from quasars, the bunching of traffic on motorways, heartbeat pulse patterns, earthquake tremors, and even the way people serve the Internet.


pg 85

Power Laws

The defining mark of both fractals and systems that display 1/f noise is the power law that expresses the relationship between events or structures of different magnitudes. If we go back to our ferromagnet hovering at the critical point and work out the size of magnetic clouds with the same orientation, we find a power law relationship between them. Small clouds are common, medium-sized ones less so, and large clouds are rare. There is, however, a characteristic ratio at work that dictates the number of large versus small clouds.

A power law is mathematician's way of saying that one measure N (e.g. the number of clouds) can be expresses the power of another quantity s (e.g. size of the clouds). Drawing the logarithmic plot of the number of clouds against the size of the clouds would reveal a straight line.

A power law is a purely statistical relationship. It will not tell you what will happen next, i.e. it cannot be used to predict whether a big quake will follow a small one or when a large stock market crash is on the cards. All it can do is tell you the distribution of events of all sizes.

Power laws can be found in the size of avalanches sweeping down the side of a sand pile. The pattern of avalanches looks the same across all time scales. The sand pile model was first introduced by Per Bak. It contained a profound lesson for anyone interested in the dynamics of self-similar systems.

Systems made up of thousands, if not millions, of interacting units managed to maintain themselves along the critical boundary. This poised state comes about purely through the interactions between the elements of the system, be they sand grains, stockbrokers or snowflakes. At the critical point it does not matter.

The interaction drives a system to self-organise. Change when it comes, comes swiftly. Once it had passed, the system settles into its old critical state. They gave the theory the name of self-organised criticality.

By going back to the sand pile we can see how this happens. Adding grains of sand drives the pile towards its critical state. When the weight of grains we are adding becomes too much for those below an avalanche occurs. Once the change has swept across the system must and settles down again into its critical state. By adding more sand we drive it again until another avalanche occurs. Between avalanches the sand is hovering around its critical point. It has become self-organised around its critical point, poised on the cusp of change and ready to react.

Bak argued that this simple model could be used to explain the origin of all the systems where 1/f noise and fractals were apparent. At the critical point the properties of the individual elements cease to matter and the interactions take over. Order emerges. Life begins.

 

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