Chaos and Complexity
Sean Pitman M.D.
Most of the the following comments are taken from a very interesting paper written by Michel Baranger from the Center for Theoretical Physics at Cambridge concerning the topics of chaos, complexity, and entropy. According to Baranger's paper chaos and complexity are very different things - although many seem to confuse or equate these two concepts.
"Chaos is very different from complexity." There is a very important facet of chaos. "[Chaos] can be, and often is, a property of very simple systems." As it turns out, "Simple questions usually have complicated answers. You ask a simple question, by giving some simple equations of motion and some simple initial conditions, and the answer to your question over time, which is the trajectory in phase space, turns out to be . . . chaotic!" Chaos, unlike calculus, is a result of "non-linear systems". However, "this does not mean that all chaotic systems are complex; far from it! . . . For one thing, chaoticity does happen with very few constituents; complexity does not. In fact, there is an enormous difference between complexity and chaos."
Not only do complex systems contain many constituents interacting nonlinearly, "the constituents of a complex system are interdependent." For example, "Consider first a non-complex system with many constituents - say a gas in a container. Take away 10% of its constituents, which are its molecules. What happens? Nothing very dramatic! The pressure changes a little, or the volume, or the temperature; or all of them. But on the whole, the final gas looks and behaves much like the original gas. Now do the same experiment with a complex system. Take a human body and take away 10%: let's just cut off a leg! The result will be rather more spectacular than for the gas. I leave the scenario up to you. And yet, it's not even the head that I proposed to cut off."
For a container of gas or a "weather pattern", no particular molecule or set of molecules is any more important to the overall function of the "system" than any other set of molecules. Complex systems are not like such chaotic systems. In fact, the workings of complex systems are more predictable than the workings of chaotic systems, even if the chaotic system is based on a rather "simple question" or collection of simple variables.
Also, a complex system possesses a structure spanning several "scales" or levels of functional complexity. For example, consider the human body again:
Scale 1: head, trunk, limbs . . .
Scale 2: bones, muscles, organs, nerves, blood . . .
Scale 3: cells, mitochondria, cytoplasm, nucleus . . .
Scale 4: chromosomes, specialized proteins (each with a special role) . . .
Note that at every scale we find a fairly static structure that must maintain its shape and relationship to other structures in order for a particular higher or "emergent" function to occur. This is an essential and radically new aspect of a complex system, and it leads to another property of complexity - "emerging" behavior.
Now, the sense in which the terms "emerging or emergent" are used here are very different from the way some people understand and use these terms. They do not denote "new" functions or properties developing over time, but the existence of collective abilities that are the result of multiple parts or processes working together at the same time in a very specified way. "Emergence happens when you switch the focus of attention from one scale to the next coarser scale above it." A certain function or behavior observed at a certain scale or "level" of function, is said to be emergent if it cannot be understood when you study it individually, without considering all the parts at that "scale" at the same time. Thus, an emerging function or behavior is a collective phenomenon unique to the scale considered that is the "result of the global interactions between the scale's constituents." For example, "The human body is capable of walking. This is an emerging property of the highest scale mentioned above (Scale 1). If you only study the head, or only the trunk, or only the leg, you will never understand walking." All the parts must be considered together in order to understand walking. The function of "walking" is in fact dependent upon all of these parts working in a very specific, predictable, and repeatable way.
"The combination of structure and emergence leads to self-organization." In other words, a pre-formed organization that works together in a specified, reproducible way, leads to an organized pattern of action - such as walking.
"Lets return now to the relationship between complexity and chaos. They are not the same thing."
"When you look at an elementary mathematical fractal, it may seem to you very 'complex', but this is not the same meaning of complex as when saying 'complex systems'. The simple fractal is *chaotic*, it is not complex. Another example would be the simple gas mentioned earlier: it is highly chaotic, but it is not complex in the present sense. We already saw that complexity and chaos have in common the property of nonlinearity. Since practically every nonlinear system is chaotic some of the time, this means that complexity implies the presence of chaos. But the reverse is not true. Chaos is a very big subject. There are many technical papers. Many theorems have been proved. But complexity is much, much bigger. It contains lots of ideas, which have nothing to do with chaos. Chaos is basically pure mathematics, and by now it is fairly well-known. Complexity is almost totally unknown still. It is not really math. It is more like theoretical physics, or theoretical anything. Of course, once it is in good shape, it will use a lot of math, perhaps a lot of new math."
"So the field of chaos is a very small sub-field of the field of complexity. Perhaps the most striking difference between the two is the following. A complex system always has several scales. While chaos may reign on scale n, the coarser scale above it (scale n - 1) may be self-organizing [i.e., greater than the sum of its parts], which in a sense is the opposite of chaos. Therefore, let us add [another] item to the list of the properties of complex systems: Complexity involves an interplay between chaos and non-chaos." In fact, many people have suggested "that complexity occurs 'at the edge of chaos', but no one has been able to make this totally clear."
In any case, it is an error to compare the fractal-type chaos of weather systems, fluid turbulence, and mountain ranges to the complexity of Boeing 747s, automobiles, computers, engines, flagellums, and higher "scales" or "levels" of complex functions found in various life forms. This is simply the comparison of apples and oranges. Their respective "complexities" are not the same thing.
Chaos, Complexity, and Entropy
A physics talk for non-physicists
Michel Baranger Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology, Cambridge, MA 02139, USA and New England Complex Systems Institute, Cambridge, MA 02138, USAMIT-CTP-3112
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