I would like to understand if it is possible to perform an experiment, where a bunch of classical harmonic oscillators (e.g., LC circuits or mechanical pendula) coupled in a simple manner (e.g., one dimensional chain with nearest neighbour coupling) act as a statistical-mechanics system with a well-defined temperature. That is, the modes of the system would be occupied according to a thermal distribution.

The crux of the problem is that small non-linearities result in an almost-integrable system, as in the Fermi-Pasta-Ulam-Tsingou exercise, which does not thermalize in a reasonable amount of time, therefore preventing a simple implementation.

I want to emphasize that this is a question about a closed system, and not some system coupled to a bath as that thermalizes readily.

To rephrase, is there a model of a chain of oscillators that I can realize on the top of my table and observe thermalization in? Any references or comments are highly appreciated!

  • $\begingroup$ Related: Slow thermal equilibrium. $\endgroup$ Commented Apr 22, 2015 at 11:04
  • $\begingroup$ So you're looking for a system to experiment on? It's very hard to make a system that is both macroscopic and sufficiently closed. (It's hard enough to make a system that's microscopic and closed.) The system would have to thermalize internally before it thermalized with the outside world. I think you'd want resonators with a high Q factor (meaning that they loose energy slowly to their environment), and the hierarchy seems to be quartz crystals (like in watches) > pendulums > tuning forks > LC circuits. (Google around.) $\endgroup$
    – lnmaurer
    Commented Apr 23, 2015 at 2:54
  • 1
    $\begingroup$ Thank you, Emilio and Inmaurer, for your comments! However, I would like to clarify that by temperature I certainly do not mean the ambient temperature. On the contrary, I mean the temperature as defined by fitting the Boltzmann distribution to the occupation of the modes of the system. I.e., if all of the oscillators are stationary, the temperature is zero. Inmaurer has correctly pointed out that a high Q factor is required, but I was hoping to come up with a system that thermalizes in, say, 100 oscillation periods. Then even the garden-variety oscillators have a sufficient Q factor. $\endgroup$
    – jarm
    Commented Apr 23, 2015 at 8:12
  • $\begingroup$ The other "stable" state (besides stationary), is if the oscillators have the "same" frequency. This state is achievable with coupled oscillators that have a "close enough" frequency. Obviously, they will "lock" to the same frequency faster, if their individual frequencies are "almost the same" and/or the coupling is "stronger." $\endgroup$
    – Guill
    Commented Apr 29, 2015 at 22:59
  • $\begingroup$ With electronic components you can create almost any level of "non-linearity" that you like (certainly in the range of 1e-6 to the order of 1) and you can easily achieve Q-factors of >1000, so I don't see why this should be a problem. I also fail to see what can be learned from such a system? Low dimensional systems are best described with chaos theory for a reason and even high dimensional systems with high Q components can be made to show stochastic resonance, so your very choice of high Q prevents the thermalization that you are looking for. $\endgroup$
    – CuriousOne
    Commented Apr 30, 2015 at 11:06

1 Answer 1


Perhaps metronomes on a movable table will match what you're looking for? Despite the two dimensional layout, it's basically a one dimensional setup.



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