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So $T$ is defined as

$$T = \left(\frac{\partial E}{\partial S}\right)$$

and $S$ is defined as

$$S = k_B \ln \Omega$$

where $\Omega$ is the number of accessible states of the system for a given $E$. I suppose you could write it as $\Omega(E)$. Hence, for $T$ to be zero, it must be possible to prepare a system that has an equivalent number of possible $|\psi\rangle_E$ for different $E$. It seems to me like you could find some weird system for which this applies. Why can't this be true?

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    $\begingroup$ David Z is giving the thermodynamic (work, internal energy, entropy, etc) approach to temperature while anna v is going by kinetic (particle velocity) theory $\endgroup$
    – gregsan
    Oct 21, 2013 at 4:05
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/32830/2451 and links therein. $\endgroup$
    – Qmechanic
    Oct 21, 2013 at 8:33
  • $\begingroup$ The condition you mention specifically (where $\Omega$ is constant over some energy range) actually leads to an infinite value of your quantity $T$, rather than zero. (By the way, Gibbs gave very good reasons why the multiplicity equations you mention do not give a robust definition of temperature, except for some special cases. So, be careful about associating them with temperature in general.) $\endgroup$
    – Nanite
    Nov 11, 2013 at 23:22
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    $\begingroup$ Possible duplicate of Is it theoretically possible to reach $0$ Kelvin? $\endgroup$
    – valerio
    Feb 9, 2018 at 12:44

5 Answers 5

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Actually, temperature is defined as

$$\frac{1}{T} = \frac{\partial S}{\partial E} = \frac{k_B}{\Omega}\frac{\partial\Omega}{\partial E}$$

So in order to have zero temperature, you would need a system with either zero multiplicity, which you can't have by definition, or an infinite derivative $\partial\Omega/\partial E$ even though the multiplicity itself is finite. Mathematically speaking, functions like that do exist, but as far as I know they haven't been found to arise in physics.

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A convenient operational definition of temperature is that it is a measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules.

temperature

The underlying framework of all matter is quantum mechanical. This means that the Heisenberg Uncertainty principle holds. Even for a single particle the HUP means that the kinetic energy cannot be constrained to zero, even more so for the ensemble of particles that compose any matter. There will always be some kinetic energy due to the HUP in any ensemble of particles.

So the reason matter cannot reach zero absolute temperatures is quantum mechanics, in my opinion.

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  • $\begingroup$ But this is using a QM description to describe a classical definition of temperature... $\endgroup$
    – Nick
    Oct 21, 2013 at 6:34
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    $\begingroup$ @Nick Physics is continuous. Classical emerges from Quantum. Quantum mechanics is necessary when one approaches limits, in many other classical variables too. This is one of them, imo. $\endgroup$
    – anna v
    Oct 21, 2013 at 6:59
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    $\begingroup$ Another example is black body radiation when the frequency distribution classically goes to infinity ( infrared catastrophy) Quantum mechanics came to the rescue there too. $\endgroup$
    – anna v
    Oct 21, 2013 at 7:04
  • $\begingroup$ I think you mean the ultraviolet catastrophy. :P $\endgroup$ Oct 24, 2013 at 20:30
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    $\begingroup$ @ÉmileJetzer yes, ofcourse :( . I had been thinking about infrared on another comment, and age shows . $\endgroup$
    – anna v
    Oct 25, 2013 at 6:22
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The third law of thermodynamics states that a quantum system has absolute zero temperature if and only if its entropy is zero: meaning it is not reacting with anything including its environment, which is impossible to achieve.

In the QM sense, 0K would be achieved when all the motion of all the particles comprising matter stops and everything comes to a standstill. BUT, the uncertainty principle says that molecules cannot stay still and continue to have a precise position. The kinetic energy cannot be limited to zero for any system, even in theory.

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Temperature is a a measure of the distribution of energy states in a system. If a system is in a pure state defined by its ground state energy then its temperature is zero even if there is a residual ground state energy as in a quantum harmonic oscillator.

For example a perfect crystal has a lowest energy ground state and energy levels are discrete so the next energy level up is separated by an energy gap. From a theoretical point of view you can imagine it being in this ground state and therefore having zero temperature.

The problem is, how could you maintain it in this state in practice assuming you could obtain it in the first place? It would have to exist in some isolated container to shield it from photons which could excite a higher energy mode. You would have to ensure that the energy of all photons that it could interact with have an energy less than the energy required to lift it to its next energy level. The problem is that the container surrounding it would also have a temperature so it could emit photons. You must ensure that it cannot emit a photon that would have enough energy to spoil the ground state of the crystal. The radiation would itself have a temperature which always means that there is a small probability of a photon with a higher energy.

In practice any low temperature system is kept within a series of insulating containers that try to isolate it from the outside would where the temperature is higher. No matter how well this is done you cannot eliminate the possibility that a photon from outside will energize a series of photons penetrating each system and exciting the crystal to a finite temperature. Temperature is defined in the context of statistical physics where even the probability of system having a raised energy level implies a non-zero temperature.

This means that you can only know that its temperature is zero if you have made a quantum measurement that puts it in a pure ground state but no measurement is that perfect unless the measurement apparatus itself has zero temperature. Furthermore immediately after the measurement there is a probability that its energy level has risen again because of an interaction from outside.

In conclusion, you can envisage a system at a zero temperature theoretically but in practice it cannot be achieved.

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  • $\begingroup$ What about a BEC? It requires a macroscropic amount of energy to transition to the next energy level. $\endgroup$
    – Nick
    Oct 21, 2013 at 17:49
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To get down to 0 k, you would need to remove All of the "heat energy" in the subject body. How would you do that, without having a "heat sink" that was already at a lower Temperature ?? Seems to me to be a second law constraint.

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