Nonequilibrium entropy, Lyapounov variables, and ergodic properties of classical systems

We discuss the problem of defining (nonequilibrium) entropy in terms of the concepts of mechanics and of reconciling its monotonic increase with the Hamiltonian evolution of the dynamical system. This leads to investigating necessary and sufficient conditions for the existence of monotonically increasing quantities or the so-called Lyapounov variables of classical systems. It is found that the condition of "mixing" is necessary and the property of being K-flow is sufficient for the existence of a Lyapounov variable. The significance of the study of Lyapounov variables for the elucidation of the fundamental questions of statistical mechanics is briefly discussed. It is seen that every Lyapounov variable must fail to commute with at least some of the operators of multiplication by phase space functions. The uncertainty relations implied by this necessary noncommutativity would then set a limit on the simultaneous determination of entropy and trajectories in phase space. These considerations thus support and sharpen the view that the thermodynamical and the (microscopic) dynamical descriptions of classical systems could be consistently reconciled as being complementary descriptions analogous to the complementary descriptions encountered in quantum mechanics.