Proof of Time's Arrow with Perfectly Chaotic Superdiffusion

The problem of Time's Arrow is rigorously solved in a certain microscopic system associated with a Hamiltonian using only information about the microscopic system. This microscopic system obeys an equation with time reversal symmetry. In detail, we prove that a symplectic map with time reversal symmetry is an Anosov diffeomorphism. This result guarantees that any initial density function defined except for a zero volume set converges to the unique invariant density (uniform distribution) in the sense of mixing. In addition, we discover that there is a mathematical structure which connects Time's Arrow (Anosov diffeomorphism) with superdiffusion in our system. In particular, the mechanism of this superdiffusion in our system is different from those previously found.