The Boltzmann Entropy and Randomness Tests

In the context of the dynamical systems of classical mechanics, we introduce two new notions called “algorithmic fine-grain and coarse-grain entropy”. The fine-grain algorithmic entropy is, on the one hand, a simple variant of the randomness tests of Martin-L¨of (and others) and is, on the other hand, a connecting link between description (Kolmogorov) complexity, Gibbs entropy and Boltzmann entropy. The coarse-grain entropy is a slight correction to Boltzmann’s coarse-grain entropy. Its main advantage is its less partition-dependence, due to the fact that algorithmic entropies for different coarse-grainings are approximations of one and the same fine-grain entropy. It has the desirable properties of Boltzmann entropy in a wider range of systems, including those of interest in the “thermodynamics of computation”. It also helps explaining the behavior of some unusual spin systems arising from cellular automata.