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Consider a two-dimensional fluid flow, confined to a square, where the bottom is held at a higher temperature than the top. With appropriate choices of the parameters, this will form a single convection cell that occupies the whole space. However, there are two possibilities - either the cell will turn clockwise or anticlockwise.

This can be seen as a (non-equilibrium) phase transition that breaks time reversal symmetry, since the density of microscopic states is concentrated into two distinct ensembles where each is the time reverse of the other. This is analogous to the way in which the Ising model spontaneously breaks up/down symmetry below its critical temperature, except that here the symmetry that gets broken is time-reversal symmetry.

However, this example is not easily amenable to analysis with statistical mechanics. Fluid flow phenomena are inherently complicated, and as far as I can see there's no simple Ising-model like abstraction of this example.

So I'm looking for a simpler example of the same phenomenon: a simple system with an explicit time-dependence, preferably with a discrete state space, that exhibits a spontaneous breaking of time reversal symmetry in the sense described above. Is there a simple standard model that's used to introduce this kind of concept?

I'm looking specifically for a model with microscopic degrees of freedom, where the time reversal symmetry is broken at the macroscopic level.

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  • $\begingroup$ I would suggest to have a look at this two papers iopscience.iop.org/0305-4470/17/12/025 link.springer.com/article/10.1140%2Fepjp%2Fi2014-14190-3 They refer to a truncated spectral decomposition of a 2D convective fluid equations.The authors end up studying a gradient type system, of which stable points represents clockwise and counterclockwise convection. Also in the following paper (in particular the reduction of the Lorenz equations to the system 2.16) iopscience.iop.org/0038-5670/21/5/R07 $\endgroup$
    – edwineveningfall
    Commented Feb 26, 2015 at 15:18
  • $\begingroup$ Practically they all are the same problem that you proposed, however they should be mathematically simpler than the original set of hydrodynamic equations. $\endgroup$
    – edwineveningfall
    Commented Feb 26, 2015 at 15:19

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I don't know if this is standard, but consider a pendulum that can swing a full circle in a plane. Vibrate the point of suspension up and down at the appropriate frequency. The pendulum will gain energy and spin either clockwise or counterclockwise.

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  • $\begingroup$ That's interesting - it clearly is an example of spontaneous time symmetry breaking, but it doesn't have the feel of a phase transition. I'll think more about this example - thank you. $\endgroup$
    – N. Virgo
    Commented Feb 24, 2015 at 7:58
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In solid state physics: Any transition to ferro- or antiferro-magnetically ordered state breaks the time inversion symmetry (1'), because the spontaneous magnetic moment on each atom changes the sign at 1'-operation.

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    $\begingroup$ Thanks for the answer. I do realise that the Ising model can already be seen as a spontaneous breaking of time symmetry for this reason, but I'm looking for a non-equilibrium transition in a system with an explicit time dependence. $\endgroup$
    – N. Virgo
    Commented Feb 24, 2015 at 8:57

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