Microscopic Dynamical Entropy I: Quantifying Hamiltonian Irreversibility in Large and Small Systems

What the math gives to ML

The paper constructs an entropy of a subsystem marginal by discarding the environment's conditional microscopic information while retaining a state-dependent conditional phase-space volume. The transferable asset is not Hamiltonian mechanics itself, but the resulting entropy functional: ordinary marginal entropy is corrected by a learned or estimated measure of how much unresolved environment volume is compatible with each representation. This suggests a representation-learning regularizer for latent-variable models that rewards informative, dynamically plausible latents without forcing the latent marginal toward a uniform distribution. Monotonicity is guaranteed only under physical assumptions such as timescale separation, so an ML implementation should treat the quantity as a regularizer or diagnostic rather than assume it is always increasing.