# Negative Temperature and bounded Energy of the Universe

I have just learned about the pretty cool idea about negative temperature (although it is "hotter" then positive temperature...) of a two level system. The condition for having negative temperature is the energy of the system has to have upper bound. It left me wondering :

If the observation of positive temperature ~2K of microwave background radiation suggests that our universe' total energy is unbounded? Are they anyway possibly linked? How do we measure negative temperature (if it were the case)?

The fact that the cosmic microwave background has a nearly perfect blackbody distribution does not rule out the possibility that the universe has an upper-bounded energy, whatever "energy" is supposed to mean in cosmology. To avoid ambiguities in the meaning of "energy," we can say it like this: The fact that the cosmic microwave background has a nearly perfect blackbody distribution does not rule out the possibility that the universe has only a finite number $$N$$ of distinct possible microstates. If you prefer, you can replace "finite number $$N$$ of distinct possible microstates" with "$$N$$-dimensional Hilbert space."

The statement that positive temperature is consistent with a finite $$N$$ is, of course, only an approximation. But it's an excellent approximation if $$N$$ is large enough. If $$N$$ is large enough, and if the temperature is close enough to zero, then the distribution will look so much like a blackbody distribution that we could not possibly ever tell the difference.

Think of a number like $$N\sim 10^{10^{1000}}$$. Finite can be practically infinite.

This is roughly analogous to the statement that even though a function in a periodic space can't be exactly a Gaussian function $$\exp(-x^2)$$, because that function is not periodic, it can still be a Gaussian function as far as we can ever hope to tell, as long as the period is much, much greater than 1.

This is important, because the universe appears to have a slightly positive value of the cosmological constant, and there are good reasons to think that in quantum physics, a universe with a positive cosmological constant must be described by a finite-dimensional Hilbert space (ref 1). This is not at all in conflict with the observation of a nearly perfect blackbody spectrum (with positive temperature) for the cosmic background radiation.

By the way, in an $$N$$-dimensional Hilbert space with large $$N$$, states that look essentially thermal as far as we can tell locally are not rare at all (ref 2).

How do we measure negative temperature (if it were the case)?

If the temperature were negative, you wouldn't need to measure it. We would all be gone. Negative temperature is hotter than any positive temperature. This makes more sense when we use the inverse temperature $$\beta\equiv 1/kT$$ instead, because this is the parameter that shows up in the usual thermal distribution $$\exp(-\beta E)$$. The limite $$\beta\to +\infty$$ corresponds to what we call "absolute zero" (the coldest situation possible), and $$\beta\to -\infty$$ corresponds to the hottest situation possible. Negative values of $$\beta$$ make sense as long as the energy $$E$$ has an upper bound, so that $$\exp(-\beta E)$$ remains finite for all allowed values of $$E$$. If $$E$$ does not have an upper bound, then $$\beta$$ cannot be negative, and $$\beta=0$$ becomes the hottest temperature possible ($$T=\infty$$).

References:

1. Witten, "Quantum Gravity In De Sitter Space" (https://arxiv.org/abs/hep-th/0106109)

2. Linden et al. “Quantum mechanical evolution towards thermal equilibrium” (http://arxiv.org/abs/0812.2385)

• thanks for the answer! btw, I did a little treasure hurting on your reference. do you think the co-author's (along with others)'s work on "Entanglement and the foundations of statistical mechanics" plausible? They claimed, ".... the main postulate of statistical mechanics, the equal a priori probability postulate, should be abandoned as misleading and unnecessary...... it should be replaced by a general canonical principle, whose physical content is fundamentally different from the postulate it replaces: it refers to individual states, rather than to ensemble or time averages. " Mar 26, 2021 at 15:43
• @Shing I think their words are a bit too strong. Assigning equal a priori probabilities is appropriate when we don't have any information that would lead us to favor one possibility over another. That remains true even if most pure states become locally indistinguishable from a thermal ensemble, which is what the paper shows. The basic rules of probability theory aren't made obsolete by new insights in quantum physics, however interesting and important those new insights might be. Mar 26, 2021 at 22:14

Normal positive temperatures occur in systems like a collection of atoms. At low temperature they are all in the vibrational ground state. If you add a little energy, some can occupy excited states. There are more ways to occupy theses states. The number of available states grows as the temperature increases. If $$S = k_B \space log(\Omega)$$ and $$T = \partial U/\partial S$$, then $$T > 0.$$

Negative temperatures are for a particular kind of unusual system, such as a collection of atoms with nuclear spins in a magnetic field. The energy is lower when spins are aligned with the field and higher when against it. At low temperature, all the spins are aligned. As the temperature rises, some spins are flipped against the field. There are more ways to occupy available states. Again $$T > 0$$.

There will be a point where about half the spins are against the field. Adding more energy doesn't change $$S$$ very much. The temperature is very large.

It is possible to add energy by other means, such as a pulse radio frequency photons. This is done in nuclear magnetic resonance. It can be arranged so that most spins are against the field. After the pulse is turned off, it can take time for the spins to decay.

In the meantime, random motion will change the spins, flipping some against and some with the field. The changes affect $$U$$, the total energy of the spins. If $$U$$ goes up, the spins would become more perfectly aligned. The available number of ways that sins can occupy these states would drop. $$S = k_B \space log(\Omega)$$ would drop. That is, disorder would drop. And therefore $$T = \partial U/\partial S < 0$$.

In this case, random changes drive the system toward increased $$S$$, and lower $$U$$. That is back toward infinite temperatures, and then into the region of positive temperatures.

This has ignored the normal vibrational states of the atoms. These are not much affected by the spins of the nuclei. Their temperature stayed positive through the RF disturbance to the nuclei and their subsequent relaxation. Negative temperatures often affect one part of a system that is weakly coupled to the rest.

The universe is not one of these special systems. The temperature of the universe is positive.