Abstract

Statistical mechanics seeks to derive macroscopic thermodynamics from microscopic dynamics, yet its central quantity— entropy— has long lacked a formulation that directly matches the thermodynamic second law. The second law of thermodynamics predicts increasing macroscopic thermal entropy, while the microscopic Gibbs entropy is conserved under Hamiltonian dynamics. To resolve this discrepancy, we introduce the Microscopic Dynamical Entropy (MDE) for a system X coupled to a bath Y , with the composite X + Y evolving under exact Hamiltonian dynamics. The MDE directly encodes the thermodynamic identity T∆S = ∆Q, and coincides with the Gibbs entropy when the bath distribution is taken to be uniform on the energy shell. In this formulation, the thermodynamic entropy increase arises from discarding information into the bath’s degrees of freedom. The MDE preserves dynamics while establishing the second law and resolving the echo paradox. We demonstrate its consistency with both classical and stochastic thermodynamics, and show explicitly that finite baths in the Zwanzig model already yield a monotonic increase of the MDE , demonstrating that irreversibility does not rely on the singular N → ∞ limit.