Action principle and Jaynes' guess method

A path information is defined in connection with the probability distribution of paths of nonequilibrium hamiltonian systems moving in phase space from an initial cell to different final cells. On the basis of the assumption that these paths are physically characterized by their action, we show that the maximum path information leads to an exponential probability distribution of action which implies that the most probable paths are just the paths of stationary action. We also show that the averaged (over initial conditions) path information between an initial cell and all the possible final cells can be related to the entropy change defined with natural invariant measures for dynamical systems. Hence the principle of maximum path information suggests maximum entropy and entropy change which, in other words, is just an application of the action principle of classical mechanics to the cases of stochastic or instable dynamics.