## 2014-01-24

### FOLLOW-UP: Gibbs Entropy and Two-Level Systems

The first professor agreed for the most part with what I had to say. He agreed that all of statistics, and therefore statistical mechanics, works only when $N$ is sufficiently large so that $\lim_{N \to \infty} \frac{\sqrt{N}}{N} = 0$ and $\lim_{N \to \infty} \frac{\ln{N}}{N} = 0$. He also pointed out that in fact there's another more specific issue, in that the derivation of the temperature of a quantum harmonic oscillator would apparently yield different ground state temperatures for different oscillator frequencies, when in fact if every oscillator (regardless of frequency) is in its ground state, the temperature should be exactly at absolute zero. Moreover, the negative Boltzmann temperature, when brought into contact with a positive temperature system, gives off heat spontaneously to the "higher" temperature system, appearing to violate the second law of thermodynamics; he showed that using Gibbs temperature instead does not in fact alleviate the problem, because the Gibbs temperature would be arbitrarily close to zero and would still give off heat to a hotter body. Finally, as I said, he also said that the experimental consequences of Gibbs temperature are not as readily apparent as the paper authors would seem to suggest.