Entropy and irreversibility in dynamical systems.

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann's principle S = klog W, does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold's 'cat' map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.