Why are simple models of coupled pendulums incapable of describing irreversible energy dissipation?

Physics Asked on January 6, 2022

Consider two pendulums $$A$$ and $$B$$ coupled by a spring and also regard $$A+B$$ to be a completely isolated system. Let us start the system in an initial configuration where only one of the pendulums (say, $$A$$) is displaced keeping the other (i.e. $$B$$) at rest. As time passes, all the energy of $$A$$ is transferred to $$B$$ that was initially at rest. At one point $$A$$ comes to rest, momentarily and $$B$$ carries all the energy. Eventually, the situation reverses and the cycle continues back and forth.

In this very simple example, if we regard one of the pendulums (say, $$A$$) to be our system of interest and $$B$$ acting as its environment, we see that there is no permanent dissipation of energy of $$A$$. Even if we track the motion of $$A$$ only, its energy is not irreversibly lost to $$B$$ (its environment). It regains all of its energy periodically without any damping.

So this simple example above illustrates that just diving a compound system into two parts and keeping track of the degrees of freedom of one part only will not necessarily lead to dissipation.

The example above could be extended to a system of $$n$$ coupled oscillators. Here too, starting with a similar initial configuration, if we regard one of the oscillators to be the system, and all the other acting as its environment, the energy of A is not irreversibly lost to the remaining $$(n-1)$$ oscillators.

Question Why do these models described above fail to capture the phenomenon of dissipation? What are the feature of the environment of an actual pendulum (i.e. a real pendulum immersed in the atmosphere) and its coupling that causes its energy to be irreversibly lost to the environment (the atmosphere)?

In a model with a finite-dimensional Hilbert space, however large, energy cannot be irreversibly lost from one part of the system. However, if the Hilbert space has $$10^{10000000000}$$ dimensions, then the local energy-loss can be close enough to irreversible for all practical purposes.

The atmosphere can (and does) emit electromagnetic radiation. Some of that electromagnetic radiation escapes into space. The electromagnetic field can be viewed as a large number of oscillators, a few for each point in space. That's uncountably many oscillators if we model space as a continuum, but even if we model space as discrete, space is pretty big, so that's still a lot of oscillators. Maybe not $$10^{10000000000}$$ of them, but easily enough to make the recurrence time longer than anything we could ever hope to observe.

That's a practical answer. Regarding questions of principle, this paper has some thought-provoking things to say about recurrence times in a cosmological setting.

Answered by Chiral Anomaly on January 6, 2022

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