  This study presents a specific symplectic map, derived from a Hamiltonian, as
a model that exhibits time-reversal symmetry on a microscopic scale. Based on
the analysis, any initial density function, defined almost everywhere,
converges to a uniform distribution in terms of mixing (irreversible behavior)
on a macroscopic level. Furthermore, we established that this mixing invariant
measure is a unique equilibrium state, unique SRB measure, and physical
measure. Additionally, through analytical proof, we have shown that the
Kolmogorov-Sinai entropy, representing the average information gain per unit
time is positive. This was achieved by validating the Pesin's formula and
demonstrating that the critical exponent of the Lyapunov exponent is $1/2$.
