Computability of entropy and information in classical Hamiltonian systems

TL;DR

This paper investigates the limits of computability of entropy and information in classical Hamiltonian systems, revealing that information capacity can grow faster than any recursive function despite initial computability.

Contribution

It introduces a formal framework for the computability of entropy and information in Hamiltonian systems and demonstrates potential noncomputability of their evolution over time.

Findings

Information capacity growth can surpass any recursive function.

Initial distributions and entropy are computable, but their evolution may not be.

Conservation of information does not imply computability of its evolution.

Abstract

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition.