Is entropy lower at zero temperature or the Planck temperature?

Question

I have already asked a question about entropy, Entropy and temperature, but I think this question is more pertinent and very different.

It is said that object with very low entropy have a lot of energy or information (ex: a UV photon has more useful energy that a visible light photon)${}^{\dagger}$. It is also said that an object that is cooled down to very low temperatures close to $-273.15 \sideset{^\circ}{}{\mathrm{C}}$ has a minimal entropy meaning that it has a low entropy.

So the question is: Which has a lower entropy:

1. $1.416808(33) \times {10}^{32} \, \mathrm{K}$ (the Planck temperature), or

2. $0 \, \mathrm{K}$ (absolute-zero temperature)?

I am asking this because I feel that the definition of entropy saying that low entropy corresponds to high energy and useful information (high-energy objects) but does not fit with the statement that cooled objects have a low entropy too. This seems kind of paradoxical to me.

${}^{\dagger}$ I might be wrong talking about the entropy as high energy, it is just that in one of by book written by Sean Carrol he states that photons with high energy have low entropy because they carry useful information to their environment however low-entropy cooled objects do not carry useful 'free energy' to their environment so why are their considered low-entropy...

Answer 1

Entropy measures the disorderness or randomness of a system. It is NOT directly related to temperature or energy, and in fact they are independent thermodynamic variables.

1. Consider a two-level system $E_1,E_2$ with $E_1<E_2$. If the $T$ is very small, then the particle is more likely to stay on the lower level 1. In the limit of $T=0$, it will for sure be on level 1. There is no ambiguity or ignorance as far as the state of the particle is concerned. This is a low entropy case. If, however, $T$ is high, then the particle will have some probability of moving up to level 2 due to thermal agitation. That's why in this case, we can not say for sure which state the particle will be in. This is ignorance, loss of information or randomness, meaning higher entropy.

2. Now considering a SINGLE level system, i.e. the system will always be in the one and only one state available. There is no randomness here, regardless of whether $E=0$ or $\infty$, $T=0$ or $\infty$. The entropy is always low.