The Hamiltonian evolution of an isolated classical system is reversible, yet the second law of thermodynamics states that its entropy can only increase. This has confounded attempts to identify a `Microscopic Dynamical Entropy' (MDE), by which we mean an entropy computable from the system's evolving phase-space density $\rho(t)$, that equates { \em quantitatively} to its thermodynamic entropy $S^{ \rm th}(t)$, both within and beyond equilibrium. Specifically, under Hamiltonian dynamics the Gibbs entropy of $\rho$ is conserved in time; those of coarse-grained approximants to $\rho$ show a second law but remain quantitatively unrelated to heat flow. Moreover coarse-graining generally destroys the Hamiltonian evolution, giving paradoxical predictions when $\rho(t)$ exactly rewinds, as it does after velocity-reversal. Here we derive the MDE for an isolated system XY in which subsystem Y acts as a heat bath for subsystem X. We allow $\rho_{ XY}(t)$ to evolve without coarse-graining, but compute its entropy by disregarding the detailed structure of $\rho_{ Y|X}$. The Gibbs entropy of the resulting phase-space density $\tilde\rho_{ XY}(t)$ comprises the MDE for the purposes of both classical and stochastic thermodynamics. The MDE obeys the second law whenever $\rho_X$ evolves independently of the details of Y, yet correctly rewinds after velocity-reversal of the full XY system.
Categories: cond-mat.stat-mech