Chapter 4 - Probabilistic chaos

Summary

Dynamical randomness and the entropy per unit time

If dynamical instability is quantitatively measured by the Lyapunov exponents, on the other hand, dynamical randomness is characterized by the entropy per unit time. The entropy per unit time is a transposition of the concept of thermodynamic entropy per unit volume from space translations to time translations. As Boltzmann showed, the entropy is the logarithm of the number of complexions, i.e., the number of microscopic states which are possible in a certain volume and under certain constraints. In the time domain, the number of complexions becomes the number of possible trajectories in a given time interval. The entropy per unit time is therefore an estimation of the rate at which the number of possible trajectories grows with the length of the time interval.

This scheme is not in contradiction with the famous Cauchy theorem which asserts the uniqueness of the trajectory issued from given initial conditions. Indeed, as in statistical mechanics, the counting proceeds with the constraint that the trajectories belong to cells of phase space. Since each cell is a continuum, the counting becomes nontrivial. Indeed, an initial cell may be stretched into a long and thin cell which will overlap several other cells at the next time step. In this way, the stretching and folding mechanism in phase space implies that the tree of possible trajectories has a number of branches which grows exponentially with a positive branching rate.

The counting may be purely topological, which yields the definition of the topological entropy per unit time of Chapter 2.