Heating is faster than cooling far from equilibrium --- and this asymmetry persists across the entire spectrum from overdamped to underdamped dynamics. An algebraic proof demonstrates that including inertial effects and phase-space relaxation does not eliminate the thermal relaxation asymmetry previously established for overdamped systems. The coupling between position and velocity degrees of freedom produces intricate relaxation trajectories that bypass local equilibria, and the velocity contribution to excess free energy does not simply vanish in the overdamped limit but depends on how temperature changes are interpreted in reduced descriptions.
The overdamped-to-underdamped transition has traditionally been treated as a simplification boundary: overdamped physics is the tractable approximation, and underdamped effects are corrections. The persistence of heating-cooling asymmetry across this boundary reveals that the asymmetry is not an overdamped artifact but a structural feature of entropy production geometry. The excess free energy decomposition shows that what appears to be a purely configurational phenomenon --- different relaxation rates for hot-to-cold versus cold-to-hot --- actually threads through momentum space as well, with the velocity degrees of freedom carrying their own irreversibility signature.
Asymmetries that survive dimensional reduction are rarely accidental. When a property persists despite the elimination of degrees of freedom that appear to carry it, the property belongs to the trajectory structure --- not to any particular subset of variables. The implication generalizes: any directional asymmetry in a dynamical system that survives coarse-graining is encoding something about the geometry of the path space that no projection can remove.
(arXiv:2603.18721)