Time-reversal symmetry vs irreversibility — flip all velocities and watch what happens
If Newton's laws are time-reversible, then for every trajectory where entropy increases, there exists a reversed trajectory (flip all velocities) where entropy decreases equally. So why does entropy always increase?
Demonstration: run the gas forward until it mixes, then flip all velocities. With no perturbation, the gas perfectly retraces its path and unmixes! Add even a tiny perturbation, and chaos amplifies it — the reversal fails. This shows irreversibility is deeply tied to sensitivity to initial conditions (chaos) and the exponential improbability of anti-thermodynamic initial conditions.