Abstract
Understanding irreversibility in macrophysics from reversible microphysics has been the holy grail in statistical physics ever since the mid-19th century. Here the central question concerns the arrow of time, which boils down to deriving macroscopic emergent irreversibility from microscopic reversible equations of motion. As suggested by Boltzmann, this irreversibility amounts to improbability (rather than impossibility) of the second-law-violating events. Later studies suggest that this improbability arises from a fractal attractor which is dynamically generated in phase space in reversible dissipative systems. However, the same mechanism seems inapplicable to reversible conservative systems, since a zero-volume fractal attractor is incompatible with the nonzero phase-space volume, which is a constant of motion due to the Liouville theorem. Here we demonstrate that in a Hamiltonian system the fractal scaling emerges transiently over an intermediate length scale. Notably, this transient fractality is unveiled by invoking the Loschmidt demon with an imperfect accuracy. Moreover, we show that irreversibility from the fractality can be evaluated by means of information theory and the fluctuation theorem. The fractality provides a unified understanding of emergent irreversibility over an intermediate time scale regardless of whether the underlying reversible dynamics is dissipative or conservative.
Highlights
Understanding emergent irreversibility in macrophysics from reversible microphysics has been the holy grail in statistical physics
In the mid-20th century, researchers revisited the issues of how entropy should be defined and why irreversibility emerges in chaotic dynamical systems [5,6]
If we formally apply the fluctuation theorem to a Hamiltonian system with no heat bath by setting the dual process to the time-reversal one that starts from the final state of the forward process, we obtain a trivial result with vanishing entropy production
Summary
Understanding emergent irreversibility in macrophysics from reversible microphysics has been the holy grail in statistical physics. One is naturally led to ask whether the key idea of the Kolmogorov-Sinai theory can justify the built-in assumption of the fluctuation theorem, thereby allowing one to evaluate emergent irreversibility and entropy production in a closed Hamiltonian system. We investigate how irreversibility emerges in a closed chaotic Hamiltonian system by incorporating a minimal accessible length scale à la Kolmogorov and Sinai. The imperfect Loschmidt demon naturally integrates the key idea of the Kolmogorov-Sinai theory with fluctuation theorems to evaluate emergent irreversibility of a closed Hamiltonian system. In Appendix B, we discuss our conjecture that the growth rate of the entropy production coincides with the Kolmogorov-Sinai entropy for a generic closed chaotic Hamiltonian system
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