Sleep involves large-scale slow rhythmic synchrony of the brain — a kind of low information entropy state because of the high degree of coherence. Despite the heat, running also involves a large-scale slow rhythmic synchrony of the body and brain. Hypothetically, high-coherence exercise might have similar cognitive effects as sleep.

This is a testable hypothesis. Simply compare low-coherence exercise (say, moving boxes or frenetic dance) with high-coherence exercise (running or rowing).