The Evolution of Complex Systems

A principle of evolution of highly complex systems is proposed. It is based on extremal properties of the information I(X, Y) characterizing two states X and Y with respect to each other, I(X,Y)= H(Y)-H(Y/X), where H(Y) is the entropy of state Y, H(Y/X) the entropy in state Y given the probability distribution P(X) and transition probabilities P(Y/X). As I(X, Y) is maximal in P(Y) but minimal in P(Y/X), the extremal properties of I(X, Y) constitute a principle superior to the maximum entropy principle while containing the latter as a special case. The principle applies to complex systems evolving with time where fundamental equations are unknown or too difficult to solve. For the case of a system evolving from X to Y it is shown that the principle predicts a canonic distribution for a state Y with a fixed average energy <E>.