# Simplest example of spontaneous breaking of time reversal symmetry

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.

• 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 Feb 26, 2015 at 15:18
• Practically they all are the same problem that you proposed, however they should be mathematically simpler than the original set of hydrodynamic equations. Feb 26, 2015 at 15:19