Entropy, oscillators and Nietzsche

I am giving two talks at a high school in Mauritius on Monday and in trying to imagine what might inspire 14 year olds, I started thinking of the cool physics videos out there (there are now also lots of nice physics games). In the wake of the Higgs discovery and the various attempts to explain it, I thought something physicists could do more often is take small and esoteric but interesting aspects of their work and try to explain it to a lay audience by using the many videos of related phenomena available, not just for publicity or to get money, but just as a natural by-product of doing research.

Actually this is a significant part of what we do anyway when we are doing research – we simplify and explain things to ourselves as we try to make sense of a new field or new results we find. One might call these Gordian challenges, after the famous story of Alexander the Great and the knot, because there are some things that seem to be almost impossible to simplify and present intuitively for a general audience, such as quantum spin.

Over breakfast this morning while flipping through facebook I saw this fun little video making the rounds again and it got me thinking again about entropy, thermalisation and reversibility. In the video a string of oscillators start together but slowly go out of phase, exhibit very complex patterns, but eventually return to their starting points. Another classic on a similar theme is this one on reversible mixing in highly viscous fluids. What both of these videos show is that the emergence of apparently chaotic, high entropy behaviour is not the same as true entropy generation, required for true thermalisation.

Like some vast meta-reference, fascinating reflections of these issues can be seen everywhere, from familiar birthday greetings (“many happy returns”) to the philosophies of antiquity and Nietzsche. Milan Kundera’s novel The Unbearable Lightness of Being begins:

“The idea of eternal return is a mysterious one, and Nietzsche has often perplexed  other philosophers with it: to think that everything recurs as we once experienced it, and that the recurrence itself recurs ad infinitum! What does this mad myth signifiy?”

I am struck by this paragraph because of the similarity of the notion of eternal return to the ideas in the videos above, and perhaps even more, to the idea of Poincaré recurrences, which play an interesting role in modern physics. Poincare recurrences, simply put, state that finite systems will eventually return arbitrarily close to their starting points (and will do so an infinite number of times). This is relevant in string theory as it attempts to understand de Sitter space, a toy model of our accelerating universe today.

In closing there is an open puzzle related to these ideas. On a rectangular, pool/snooker/billiards table, one can easily calculate under what conditions a ball fired out at some angle from a wall will eventually visit every point on the table and when it will form a repeating pattern (known as a periodic orbit). It turns out that the key thing is whether the tangent of the angle is a rational number or not.

But what about balls bouncing around a perfect triangular pool table? Can you always find a periodic, repeating, orbit, or are there triangles for which there are no periodic orbits? Turns out this is an unsolved problem, and fun to think about. So have a go, you might get famous.

The issues of entropy generation and irreversibility are particularly relevant in cosmology where there have been interesting debates on the quantum-to-classical transition in the early universe after inflation. For some it is enough to produce many particles, others are happy as long as the off-diagonal parts of the density matrix oscillate sufficiently rapidly (so they would time-average to zero) or the fluctuations are in a squeezed state. Others want to see true decoherence and to split the universe into a bath and a system.

You can see this environment interaction happening at the ends of both videos linked to above. Neither system quite returns to their starting points – they have been slightly messed up. To my knowledge this issue of entropy production in the early universe, and the intimately-related issue of the quantum-to-classical transition, has not been fully resolved.

On an intimately related but very sad note, I heard yesterday that one of the proponents of squeezed states in cosmology, and one of the first to study the observational effects of primordial gravitational waves, Leonid Grishchuk, has passed away. I first encountered Leonid when I was at a student at a school in Erice when he stood up and shouted at a famous cosmologist  “Everything you have said is wrong, and I will tell you why!” Despite his old-school Russian style, he was very warm hearted and kind. We have lost one of the great characters of cosmology.


About Bruce Bassett

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