Wednesday, 19 November 2008

Building up a list of 'quantum chaos' links.

Connections to these thoughts

- Abstractions and their significance.
- Quantum Chaos, Martin Gutzwiller, Scientific American, January 1992

A case brought forward about quantum chaos

"At about the time of Poincare's seminal work on classical chaos, Max Planck started another revolution, which would lead to the modern theory of quantum mechanics. The simple systems that Newton had studied were investigated again, but this time on the atomic scale. The quantum analogue of the humble pendulum is the laser; the flying cannonballs of the atomic world consist of beams of protons or electrons, and the rotating wheel is the spinning electron (the basis of magnetic tapes). Even the solar system itself is mirrored in each of the atoms found in the periodic table of the elements. Perhaps the single most outstanding feature of the quantum world is its smooth and wavelike nature. This feature leads to the question of how chaos makes itself felt when moving from the classical world to the quantum world. How can the extremely irregular character of classical chaos be reconciled with the smooth and wavelike nature of phenomena on the atomic scale? Does chaos exist in the quantum world'? Preliminary work seems to show that it does. Chaos is found in the distribution of energy levels of certain atomic systems; it even appears to sneak into the wave patterns associated with those levels. Chaos is also found when electrons scatter from small molecules. I must emphasize, however, that the term 'quantum chaos' serves more to describe a conundrum than to define a well-posed problem. "

No solid base? Quantum chaos not been irrevocably confirmed or in a milder version surely footed? The term only used to describe a conundrum faced up with, a curiosity but not the processes, quantum processes, themselves?

"Considering the following interpretation of the bigger picture may be helpful in coming to grips with quantum chaos. All our theoretical discussions of mechanics can be somewhat artificially divided into three compartments [see illustration] although nature recognizes none of these divisions. Elementary classical mechanics falls in the first compartment. This box contains all the nice, clean systems exhibiting simple and regular behavior, and so I shall call it R, for regular.
Also contained in R is an elaborate mathematical tool called perturbation theory which is used to calculate the effects of small interactions and extraneous disturbances, such as the influence of the sun on the moon's motion around the earth. With the help of perturbation theory, a large part of physics is understood nowadays as making relatively mild modifications of regular systems. Reality though, is much more complicated; chaotic systems lie outside the range of perturbation theory and they constitute the second compartment. Since the first detailed analyses of the systems of the second compartment were done by Poincare, I shall name this box P in his honor. It is stuffed with the chaotic dynamic systems that are the bread and butter of science. Among these systems are all the fundamental problems of mechanics, starting with three, rather than only two bodies interacting with one another, such as the earth, moon and sun, or the three atoms in the water molecule, or the three quarks in the proton. Quantum mechanics, as it has been practiced for about 90 years, belongs in the third compartment, called Q.

Mechanics, stretching that notion wider, to include anything that is described by, and, as systems such as social, mental, psychological and therefore social, mental, psychological mechanics? Attempting such an act based broadly on chaos self-similarity principle, and even further, by its virtue to go ahead and look at social, mental, psychological mechanics in novel ways?

And what about that elaborate mathematical tool called perturbation theory, which is used to calculate the effects of small interactions and extraneous disturbances to regular systems? Could it be of any use in social, mental, psychological systems? Or should we take stalk of what Henri Poincare surmised, as it is mentioned in the same website.

"So thereafter, the great French mathematician-astronomer-physicist Henri Poincare surmised that the moon's motion is only mild case of a congenital disease affecting nearly everything. In the long run Poincare realized, most dynamic systems show no discernible regularity or repetitive pattern. The behavior of even a simple system can depend so sensitively on its initial conditions that the final outcome is uncertain."

That there is, no discernible regularity or repetitive pattern, in most dynamic systems. That the behaviour of even a simple system can depend so sensitively on its initial conditions, that the final outcome is uncertain. Even a simple system? What is simple, but a construct, our minds devise, by removing all information contained in a system, or object, apart from what our minds deem as necessary? And being doing it, on and on, in individual or collective level alike? An act of abstraction that help us gain knowledge?

The necessary information, within the range defined by the maximum of our cognitive apparatus potential and the minimum needed for the overall information to make sense. What our minds can handle, but in reality systems possess a lot more information than we actually see (likened to a tip of the iceberg?), observe, and by virtue of that, no system is as simple as it looks, and therefore, it can not be predicted, as Henri Poincare surmised.

Is that a blessing or a curse? It is certainly a blessing.

So perturbation theory, dealing with the minute differences, in initial conditions. A precursor of that feature of chaos? I remember reading about mathematical calculations of hard problems, riddled with infinities, which perturbation theory have been removing. Does these menacing infinities have anything to do with the dynamics of chaotic states? In a way, describing the pull or push to the path of an unfolding trajectory? That is pulled towards infinity?

And what could we make out of the statement

"With the help of perturbation theory, a large part of physics is understood nowadays as making relatively mild modifications of regular systems."

or even

"The main connection between R and P is the Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theorem provides a powerful tool for calculating how much of the structure of a regular system survives when a small perturbation is introduced, and the theorem can thus identify perturbations that cause a regular system to undergo chaotic behaviour."

How much can a system take? When a small perturbation is introduced? A small change of rule, or norm or a habit? Referring to the quality of the perturbation. Its overall effect in a regular system or just a system. Taken in to account that from all the perturbations possible, it is bound to be, that only a few would actually have a profound effect on the structure of a regular system, the state of a system, the stable state attractor. Most of the perturbations would have a negligent effect.

Why did I think about conservative values? The attractors, which we will not want to change? Content with the status-quo? We do not want changes.

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