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Regardless of studies and debates over a century, the statistical origin ofthe second law of thermodynamics still remains illusive. One essential obstacleis the lack of a proper theoretical formalism for non-equilibrium entropy. HereI revisit the seminal ideas about non-equilibrium statistical entropy due toBoltzmann and due to Gibbs, and synthesize them into a coherent and preciseframework. Using this framework, I clarify the anthropomorphic principle ofentropy, and analyze the evolution of entropy for classical Hamiltonian systemsunder different experimental setups. I find that evolution of Boltzmann entropyobeys a Stochastic H-Theorem, which relates probability of Boltzmann entropyincreasing to that of decreasing. By contrast, the coarse-grained Gibbs entropyis monotonically increasing, if the microscopic dynamics is locally mixing, andthe initial state is a Boltzmann state. These results clarify the precisemeaning of the second law of thermodynamics for classical systems, anddemonstrate that it is the initial condition as a Boltzmann state that isultimately responsible for the arrow of time.

Definitions and Evolutions of Statistical Entropy for Hamiltonian Systems