Entropy and information in processes of self-organization: uncertainty and predictability

The mean uncertainties of probability distributions of discrete sets of events and those of dynamic sequences of events are investigated on the macroscopic and on the microscopic level. First one-time distributions are studied. The key point is the specification of the state space; the (coarse-grained) physical phase space leads to the thermodynamical entropy and the order parameter space to the informational entropy. In a similar way the uncertainty of the state after one step ahead in time, introduced by Shannon, McMillan and Khinchin, is related to the dynamic entropies studied by Kolmogorov and Sinai and to the gain of information. The non-equilibrium entropies and their relation to properties of attractors are discussed for several examples taken from physical and non-physical self-organization processes.