Jul 022000

We feel more comfortable in the natural breeze than that mechanically caused by the fan. You might consider the natural breeze capricious, but there is a rhythm of life whose order can be mathematically formulated.

1. Zipf’s law and the order of fluctuation

G.K. Zipf investigated the frequency of English words and found that the frequency of the Nth most frequently used word was 1/N the height of that of the most frequently used word, "the". The frequency of "the" is 0.1; the frequency of the second most frequently used word, "of", is 0.05, one half of that of "the"; the frequency of the third most frequently used word, "and", is 0.033, one third of that of "the", the frequency of the fourth most frequently used word, "to", is 0.025, one fourth of that of "the"; and so on. Plotting the frequency ranking and the frequency on a double-logarithmic diagram, you will find it shows a lower right straight line,

f(logN) = -(logN+log10)


y = -x-1

This is the Zipf’s law.

Zipf’s law can be applied to the relation between frequency and power spectrum of various kinds of fluctuations. Such fluctuations are called 1/f fluctuations. Natural fluctuations seem to be disorderly, but there is an order of inverse proportion: the big a power spectrum is, the less frequently it appears.

Not all fluctuations are however 1/f fluctuations. Some are 1/f2 and in this case, inclination of the line is not -1. To cover such cases the Zipf’s law is extended to the power law: y=c/fn (c and n are constants).

There are two kinds of distribution except 1/f fluctuations: white noise and monotone. You can compare three types of arranging alphabets

  1. cEjkrWiUYsvNlhaXdFmOy (white noise)
  2. This is a normal sentence. (1/f fluctuation)
  3. aaaaaaaaaaaaaaaaaaaa (monotone)

When the arrangement gets near to the first type, the difference in frequency decreases and the inclination on the double logarithm scale graph becomes horizontal. When the arrangement gets near to the second type, the difference in frequency increases and the inclination on the graph becomes perpendicular.

2. The rhythm of life is 1/f fluctuation

The science of complexity locates life at the edge of chaos. Life is neither dead disorder nor dead order, but a critical point of phase-transition from chaos to order. In other words the life principle is neither white noise nor monotone, but a 1/f fluctuation.

We are often "beside ourselves" with ecstasy over beautiful music or scenery, while we feel an aversion to high entropy such as a din or filth and weary of low entropy such as monotonous sound or figure, that is to say, we keep away from the two extremes, pure low entropy and utter high entropy. When you analyze the power spectrum of sound frequency, you will find that beautiful music shows 1/f frequency. You can also find 1/f fluctuations in curves rich in changes characteristic of beautiful natural scenery.

It is because we are complex systems with 1/f fluctuations that we feel a comfortable harmony in 1/f fluctuations. When we find ourselves in others, the complexity gap between others and ourselves disappear, so we are "beside ourselves".

It is not only men that feel pleased with 1/f fluctuations. When a poultry farmer made their hens listen to music with 1/f fluctuations, they feel relaxed and laid more eggs than usually. 1/f fluctuation is not only a human principle but also a life principle.

Let’s consider this life principle on the social level. The totalitarian state such as the Soviet Union under bureaucratic control, where the state regulates people too much, is a dead order, while anarchy such as Yugoslavia under the civil war is a death of state. The vivid social system can be realized neither by a big government nor by anarchy, but by the small government that keeps an exquisite balance between the regulation of state and the freedom of society.

3. The paradox of fluctuation

Now I bring up some questions:

Why do we consider machines inhuman, though they are made by us, while we feel friendly to nature that we did not contribute to? Why do we pursue simplicity as intelligibility, though we ourselves are complex systems and often despise simplicity? Why do we feel alien to low entropy monotone, though we are negentropy against the increase in entropy as white noise?

My answer is:

Although we are complex as a sensitive being, our understanding is essentially simple and trying to simplify the complex world makes us intermediate being. It is easy to make a simple and determinate theory, but it is difficult to apply it to the complex and indeterminate world. It is easy to hold up a simple and determinate ideal, but it is difficult to realize it in the complex and indeterminate world. It is an economic principle that the more difficult acquiring something is, the more valuable it becomes. That’s why our complex endeavor to simplify complexity is more valuable than the simple goal itself.

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