What separates classical and quantum chaos? Premium

Weather forecasting is difficult due to the unpredictable nature of the atmosphere. The 'butterfly effect' metaphor captures this notion, as small changes in conditions can have large impacts. Deterministic chaos explains why the future can only be predicted if the present is known with great accuracy. Mathematics of chaos is diverse, and quantum mechanics can be applied to study it. Rydberg atoms are a link between classical and quantum domains, and chaos in them is detected in the spectrum of energy levels. Quantum chaos is a new and exciting area of research with implications in many fields.

Consider weather forecasting. The earth’s atmosphere is a laboratory of randomness, whose conditions change frequently in terms of its pressure, density, the flow-rates of various gases, and temperature. As a result, the paths of gas molecules become increasingly unpredictable. This is why a weather phenomenon that has been predicted to last for a longer period of time is unlikely to be as accurate – or even true, for that matter – for a more intermittent duration.

This is why a simple model that tracks the way heat is moved through the atmosphere can possess a lot of unpredictability. Such notions are captured by the term ‘butterfly effect’ – named for the idea that the mere flapping of a butterfly’s wings in one place can produce a storm somewhere else. This may sound ridiculous but as a metaphor, it has a well-understood scientific basis.

The pinball machine makes for an illustrative example: the motion of the little ball is precisely governed by the laws of gravitation and motion, of rolling, elastic collisions, scattering, etc. – yet it is practically impossible to accurately predict the ball’s position at a given moment (well after someone starts playing it). Such systems are said to be classically chaotic because of their apparently unpredictable behaviour even though they are governed by deterministic physical laws.

In fact, a more appropriate term that defines such systems would be deterministic chaos. Deterministic chaos essentially means that the future can be predicted only if the present is known with a great degree of accuracy. However, if the present is known only approximately, the future can’t be predicted. This is also what the term ‘butterfly effect’ stands for: that some system is highly sensitive to its starting conditions. Even a small change in these conditions can produce disproportionately large changes in the way the system evolves.

The mathematics of chaos – as well as its applications, for that matter – is highly diverse. It incorporates the study of such systems as the turbulent flow of fluids, irregularities in the human heartbeat, irregular patterns in the amplitude of sound transmission, trends in population dynamics, voting patterns in an election, power transmission in electrical circuits, chemical reactions, the physics of the state of matter called plasma, planetary dynamics in the inner solar system, and the motion of clusters of stars.

As noted earlier, a chaotic system is very sensitive to its initial conditions and the dynamics thereof. As a result, such a system can seem to behave randomly rather than in a regular manner. The duration for which the system’s evolution will be predictable depends on a few things, such as how accurately and precisely its present state is known, the amount of uncertainty that it can tolerate, and a time factor determined by the dynamics of the system, called the Lyapunov time.

For example, in a chaotic electrical circuit, the Lyapunov time is about 1 ms. For weather systems, it is a few days, and for the inner solar system, it can be 4-5 million years.