What is the temperature of the residual gas in a dilution refrigerator? In a dilution refrigerator, the inner vacuum can is cooled by the mixing chamber to a base temperature of a few millikelvin. This experimental volume is kept under ultra high vacuum with typical pressures far below $10^{-10}$mbar. What is the effective temperature of the residual gas?
In thermodynamic equilibrium, that temperature would just be the temperature of the inner vacuum can. But that isn't right here: The residual gas pressure is established because the outgassing rate from materials equals the vacuum pump speed. Hence, there is a constant supply of atoms (mostly He and H2) outgassing from the inner vacuum can. What comes out however is the high-temperature tail of those atoms diffusing out of the vacuum can. Hence, the effective temperature of this residual gas will be well above the temperature of the vacuum can. But what is it? Can it be calculated, or has it been measured?
 A: 
This experimental volume is kept under ultra high vacuum with typical pressures below $10^{-10}$ mbar.

The vacuum level in an experimental volume mounted to a dilution refrigerator is far lower than this.  $10^{-10}$ mbar is a fairly high vacuum at room temperature (though pressures much lower than this, in the range of $10^{-12}$ mbar, are possible). But if the entire experimental volume is below 4.2 K, then the vacuum level drops far below the detection threshold of any instrument I am aware of.



*Characteristics of a cryogenic extreme high-vacuum chamber.
(USA)

We have designed, built, and tested a cryogenic chamber capable of
pressures far below $10^{-14}$ torr (~$10^{-12}$ Pa) for surface studies. In a conventional vacuum system the surface desorbing and diffusing/
desorbing vapors enter the chamber and are pumped to a dynamic
equilibrium pressure by an ion pump or a cryopumping surface. In
contrast, in the present system these vapors are prevented from first
entering the cubic-foot chamber by cooling to 4.2 K all walls and all
instruments sealed within it. This results in a nearly perfect vacuum
(~1 atom/cm$^3$), which has been used to preserve sample surfaces,
between measurements (i.e. 60 days). At higher chamber temperatures we have used an in situ quadrupole mass analyzer to measure
the equilibrium partial pressures from 300 down to $\sim$ 30 K, where
the last two gases, hydrogen and helium, drop below the instrument’s detectability limit.
W Thompson and S Hanrahan, J Vac Sci Technol, 14 (1), 1977, 643-645.

With that being said, in principle there are still a few particles bouncing around inside the experimental volume, so your question is not invalid.  However, even defining the temperature of this tenuous population of atoms is problematic.
On average, the particles in the experimental volume will collide with the walls many, many times before colliding with each other, so the particles do not come to equilibrium amongst themselves. To the extent that they can be said to be in equilibrium with anything, it would ultimately be with the walls. As high energy particles desorb from the walls, the energy in the walls decreases, but after a few collisions this energy excess is returned and the system comes to a tentative equilibrium (though it should be said that during each collision, there is a fairly high probability that the particle will stick to the wall).
