# Why must Debye cooling eventually slow down?

Masanes and Oppenheim derive a maximal cooling rate $T(t)\geq k/t^7$ in their paper on the third law of thermodynamics. This is not just a bound on passive cooling, but also active cooling using any kind of thermal machine.

Now consider a spherical blackbody object with a heat capacity set by the Debye model, $C_V=\frac{12\pi^4 N_A k_B}{5T_D^3}T^3=\alpha T^3.$ Energy balance gives $$\frac{dT}{dt}=-\frac{A(r) \sigma}{V(r)C_V(T(t))}T(t)^4 = \frac{3\sigma}{\alpha r}T(t)$$ The solution is $$T(t)=T(0) e^{-(3\sigma/\alpha r)t}.$$

This declines exponentially in temperature and will sooner or later break the Masanes and Oppenheim bound. So, what hidden assumption fails here?

The paper assumes that the heat reservoir I am cooling with has a finite (if suitably large) volume and that fluctuations are bounded. So one possibility is that the Planck radiation law implicitly assumes something about the modes that breaks these assumption (e.g. space acts as an infinitely large heat reservoir).