Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 344 / 2 / d

At $\xi =0$, the polar order parameter obeysFor $a<0$, $b>0$, the magnitude relaxes to $\sqrt{-a/b}$ while the direction rotates continuously. Thus the likely ordered behaviour is tumbling rather than a fixed alignment.

More generally, for $|\xi |<1$ and $g\ne 0$, $\xi \cos 2\theta -1$ never vanishes, so the tumbling of a polar order parameter persists. The polar order parameter completes a full $2\pi$ rotation inobtained by integrating $dt=2|d\theta |/(|g|(1-\xi \cos 2\theta ))$ over a full rotation. An unoriented nematic director repeats after half this period, whereas a polar order parameter distinguishes opposite directions. For $\xi \ne 0$ its magnitude can also oscillate. If the polar order parameter decays to zero, its orientation is no longer a physical observable; the tumbling conclusion concerns a persistent ordered state.