The Heating Asymmetry

Heat something far from equilibrium and it reaches the new temperature faster than it cools from the same distance. This asymmetry between heating and cooling has been established for overdamped systems — those where friction dominates and inertia is negligible. The question is whether the asymmetry survives when inertia matters.

It does. The proof is algebraic, covering the entire range from overdamped to underdamped dynamics.

The result extends to phase-space relaxation, where both position and velocity must reach equilibrium. Heating is still faster than cooling. The asymmetry is not an artifact of the overdamped approximation but a structural feature of how systems relax when driven far from equilibrium.

A subtlety emerges at the boundary. In the overdamped limit, velocity degrees of freedom should become irrelevant — they’re infinitely fast compared to position. But the contribution of velocity to the free energy doesn’t simply vanish. Instead, it depends on how the temperature change is interpreted in the overdamped description. The same physical process — changing the temperature of a Brownian particle — maps onto different overdamped models depending on whether you treat the temperature change as instantaneous or gradual. The phase-space description resolves this ambiguity; the overdamped description introduces it.

The asymmetry’s persistence across all damping regimes suggests it’s thermodynamic, not mechanical. It doesn’t arise from the specifics of how the system moves through phase space but from the geometry of the free-energy landscape itself. Far from equilibrium, the landscape is steeper on the heating side than the cooling side. Inertia changes the trajectory through the landscape but not its shape.