Hamiltonian Heat Baths, Coarse-Graining and Irreversibility: A Microscopic Dynamical Entropy from Classical Mechanics Article Swipe
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· 2025
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.2503.11334
· OA: W4417285500
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 $ρ(t)$, 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 $ρ$ is conserved in time; those of coarse-grained approximants to $ρ$ show a second law but remain quantitatively unrelated to heat flow. Moreover coarse-graining generally destroys the Hamiltonian evolution, giving paradoxical predictions when $ρ(t)$ 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 $ρ_{XY}(t)$ to evolve without coarse-graining, but compute its entropy by disregarding the detailed structure of $ρ_{Y|X}$. The Gibbs entropy of the resulting phase-space density $\tildeρ_{XY}(t)$ comprises the MDE for the purposes of both classical and stochastic thermodynamics. The MDE obeys the second law whenever $ρ_X$ evolves independently of the details of Y, yet correctly rewinds after velocity-reversal of the full XY system.
