Role of parity transformation for the fluctuation theorem: limit cycle and symmetry breaking

The fluctuation theorem of the Crooks type is studied for thermodynamic nonlinear-multivariate systems. In particular, a bivariate system having a limit cycle is discussed in detail. It is explicitly shown how the time reversal operation has to be combined with the parity transformation in order to derive the fluctuation theorem for the change of the "renormalized entropy." Furthermore, breakdown of the symmetry in a limit-cycle model due to the effect of fluctuations is also discussed.