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The positivity of entropy change rate was rigorously proven in a certainmicroscopic system associated with a Hamiltonian using only information aboutthe microscopic system. This microscopic system obeys an equation withtime-reversal symmetry. We proved that a symplectic map with time-reversalsymmetry has the same ergodic properties as an Anosov diffeomorphism. Thisresult guarantees that any initial density function defined except for a zerovolume set converges to the unique invariant density (uniform distribution) inthe sense of mixing. Moreover, we proved that the Kolmogorov--Sinai entropy,which is the entropy change rate, is \textit{positive} using the property ofAnosov diffeomorphisms. In addition, we discovered a mathematical structureconnecting the convergence of initial density functions to the uniformdistribution and positivity of the entropy change rate (Anosov diffeomorphism).

Proof of Irreversibility for Systems with Time Reversal Symmetry