Compression and information entropy of binary strings from the collision history of three hard balls

We investigate how to measure and define the entropy of a simple chaotic system, three hard spheres on a ring. A novel approach is presented, which does not assume the ergodic hypothesis. It consists of transforming the particles' collision history into a sequence of binary digits. We then investigate three approaches which should demonstrate the non-randomness of these collision-generated strings compared with random number generator created strings: Shannon entropy, diehard randomness tests and compression percentage. We show that the Shannon information entropy is unable to distinguish random from deterministic strings. The Diehard test performs better, but for certain mass-ratios the collision-generated strings are misidentified as random with high confidence. The zlib and bz2 compression algorithms are efficient at detecting non-randomness and low information content, with compression efficiencies that tend to 100% in the limit of infinite strings. Thus 'compression algorithm entropy' is non-extensive for this chaotic system, in marked contrast to the extensive entropy determined from phase-space integrals by assuming ergodicity.

Automated summary

We investigate how to measure and define the entropy of a simple chaotic system without assuming the ergodic hypothesis. An approach of converting collision history into binary digits is proposed. Three methods are used to demonstrate the non-randomness of collision-generated strings compared with random number generator created strings: Shannon entropy, diehard randomness tests, and compression percentage. The study shows that Shannon entropy fails to distinguish random from deterministic strings, while the Diehard test misidentifies collision-generated strings as random under certain conditions. However, the zlib and bz2 compression algorithms efficiently detect non-randomness and low information content, with compression efficiencies approaching 100% for infinite strings. Consequently, the "compression algorithm entropy" of this chaotic system is non-extensive, contradicting the extensive entropy determined by assuming ergodicity based on phase-space integrals.