Transient Fractality as a Mechanism for Emergent Irreversibility in Chaotic Hamiltonian Dynamics

Boltzmann refuted the Loschmidt irreversibility paradox by arguing that the number of events with a positive entropy production should be infinitely more than that with a negative entropy production despite the one-to-one correspondence between them. His idea was later verified by numerical simulations of reversible dissipative equations of motion, where it was confirmed that one should sample states exactly on a zero-volume fractal attractor to decrease the entropy. Thus, the probability for a negative entropy production vanishes although it is allowed by the equations of motion. We here address the question of whether this fractal picture applies to chaotic Hamiltonian dynamics. Although the Liouville theorem excludes fractality in the long-time limit, we find that a fractal structure transiently emerges in the Bunimovich billiard. Moreover, this transient fractality can be reformulated in light of the flucutuation theorem. As a result, we give a lower bound for an information-theoretic irreversibility, which is determined by the transient fractality.