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Imagine a solid box in deep space. Solids are as far as I know constructed by positive nuclei in some sort of coherent structure, with electrons orbiting nearby too.

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Both the nuclei and the electrons are charged objects, and accelerating charged objects emit electromagnetic radiation. Because of the object's temperature, the nuclei vibrate and oscillate a little bit, meaning that no matter how small, the box should still give off electromagnetic radiation. Over time, the box should lose it's internal 'heat energy', and approach a temperature of $0\text{K}$, where there is absolutely no more vibrations.

However, I know that due to quantum mechanics, electrons that 'orbit' a nucleus don't actually move, but exist in some stationary state, that emits no radiation. Does a similar reasoning apply to the nuclei? Or will they actually continue to emit radiation, and if so, can we calculate the nuclei's average kinetic energy as a function of time?

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You're right that the classical idea of radiation emission from an accelerated charge cannot be applied to electrons in orbit around nuclei, and thus they do not emit radiation (unless they're in an exited state and decay to a lower state).

The same thing does not apply to the nuclei. As you suspect, they will, over time, lose energy and vibrate less and less. The rate at which they lose energy decreases, so the temperature decrease only asymptotically approaches the ambient temperature.

So, what do I mean by the ambient temperature? Well, no matter how far away in deep space you place your box, it will be irradiated by the cosmic mirowave background (CMB), which at the moment has a temperature of 2.7 K, and thus your box won't be able to cool below this temperature.

That is, unless you're willing to wait for billions of years. Since the temperature of the CMB itself decreases, patiency can get you arbitrarily close to 0 K.

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  • $\begingroup$ fwiw it's possible for objects to cool transiently below the CMB temperature as radiative equilibrium is a time-average dynamic equilibrium, but this sort of short-lived 'undershoot' occurs with almost certainly imperceptible significance and time scales $\endgroup$ – oldrinb Nov 19 '17 at 8:56
  • $\begingroup$ @oldrinb Yes, that's completely correct. A nice example of this is the Boomerang Nebula which expands so fast that it has cooled to 1 K. In a few thousand years, though, it will have heated up to the CMB temperature. $\endgroup$ – pela Nov 19 '17 at 10:49
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To add a little detail about radiative thermal equilibrium:

As atoms at nonzero temperature collide with each other, they do emit electromagnetic radiation, and if they were in an empty universe, they would approach zero temperature. However, since the universe isn't empty, they also absorb electromagnetic radiation coming from all the other atoms around.

If your group of atoms are at a higher temperature than their surroundings, they emit more than they absorb, and cool down. If your group of atoms are at a lower temperature than their surroundings, they absorb more than they emit, and heat up. This continues until they reach equilibrium. At that point, they emit energy at the same rate that they absorb it, and their temperature stops changing.

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