Why must Debye cooling eventually slow down?

Question

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).

Answer 1

Because Debye cooling behaves the same way with time regardless of when time zero is (t = 0), the initial value of time for it is arbitrary. However, time zero corresponds to a singularity for the Masanes and Oppenheim bound.

Is time zero the moment of the big bang? Or when an object is first subjected to aggressive cooling mechanisms? This may have a profound impact on how quickly Debye cooling reaches the Masanes and Oppenheim bound.

Based on the paper, the Masanes and Oppenheim bound works as it does because we are in Euclidean three-space. Additional spatial dimensions or hyperbolic space would permit increased cooling rates.

An inverse proportion to the seventh power of time is still quite a weak bound. An object that becomes ten times older (relative to time zero) would drop in absolute temperature by a factor of ten million (10,000,000).