Phenomenological determinism in Hamiltonian dynamical systems

Abstract Classical equations that govern the information in a Boltzmann–Gibbs Hamiltonian dynamical system are shown to entail the evolution of a phenomenological fraction of the information. The evolution of this fraction of information can be predicted along a preferential time direction provided a structural condition on the dynamics time scale is fulfilled. This preferential direction points towards the future or the past, starting from an initial time, provided an appropriate initial condition is fulfilled by the Boltzmann–Gibbs distribution at this time. Choosing one direction defines an arrow of time. That fraction of information is thus submitted to a phenomenological determinism. It permanently generates non-phenomenological components of the information, which accumulate, and experience an exponentially fast complexification. This complexification makes the accumulated non-phenomenological information unable to influence the evolution of the phenomenological information. That evolution is thus autonomous and deterministic.