Nature has particle accelerators that are far beyond our capacity, but occasionally I hear atomic physicists claim that they are able to make something that has never been formed in any natural process (that is, not coming from intelligent life). This has always seemed plausible to me, but is it really true?

For definiteness, let's say we have produced 100,000 atoms at 100 nK for 10 seconds, which is quite conservative in every parameter. What is the probability of this many atoms having ever taken this temperature for this amount of time due to spontaneous fluctuations, anywhere in the observable universe? How about 10 atoms? Supposing the universe is infinite in size, what volume would we have to look at before we had a decent probability of this ever occurring? Feel free to make any simplifying assumptions that would give an upper bound to this probability, such as assuming that the universe has been at 3 K the entire time since the Big Bang.

Edit: since no one has bitten, I will point out the farthest I've gotten on this question: the fluctuation theorem states, if I understand it correctly, that following:

$\text{Pr}(\Delta S=-S_{UC})=\text{Pr}(\Delta S=+S_{UC})e^{-S_{UC}t}$

where $S_{UC}$ is the entropy decrease needed to take the atoms from 3 K to 100 nK. In other words, it is exponentially more likely that they will spontaneously increase by the needed entropy, itself presumably an unlikely event.

  • $\begingroup$ Keep working on it, and let us know what you figure out :) $\endgroup$ – Eriek Apr 15 '15 at 5:22

Although not a complete answer, one place to start is with the coldest naturally occurring place in the universe, which is the Boomerang Nebula, a planetary nebula that is around 1 K. As best as I can tell, this cooled below the CMB temperature simply by adiabatic expansion, and is insulated in its interior from CMB heating. Is this a feasible way to get to ultracold temperatures?

For a monoatomic gas, recall that adiabatic expansion is $TV^{2/3}=const$. So something that cools to 100 nK from 3 K would have to expand in volume by a factor of ~$10^{11}$. Which is, obviously, a lot, but space is big and has more than enough room. The Boomerang nebula is about 1 light-year across and is expanding out at about 164 km/s. So we could imagine, for example, a similar object that starts out with a radius of around 10^(-4) LY (which is still 10,000 times larger than the Sun) and expands to the same size at the same rate, which would take around 1000 years. This doesn't seem particularly implausible, although I'm no astronomer.

The harder question to answer is what the heating rate from the CMB would be in the interior of this cloud. It would only have to be very small, of course, to counteract the adiabatic cooling. Looking at one of the papers on the Boomerang nebula, the authors there estimate the cosmic ray heating as $4*10^{-28}$ erg/s, while the cooling rate is around $10^{-25}$ in the same units. So since adiabatic cooling goes slower as the gas gets colder (indeed, in this simple model we have $\dot{T}(t)=-T/t$), we would probably expect that by the time the gas has cooled to about 1/1000th of the CMB temperature, if not sooner, the heating rate would match the cooling rate.

All in all, my very crude best guess then is that adiabatic expansion of this sort could not lead to a temperature below the mK scale.


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