# Is there a “high temperature” variant of 0 degrees Kelvin? [duplicate]

I know that -273.15 degrees celsius, also known as Absolute Zero or 0K is the low temperature limit for objects, but is it possible that there is a 'highest temperature?'

I would have to guess that it's when it allows objects to reach light speed, but I believe there has to be some 'set temperature' that is the limit for anything. Am I wrong or is there an actual answer?

• possible duplicate of Why is there no absolute maximum temperature? – ACuriousMind Jan 12 '15 at 0:00
• @ACuriousMind would that question not be already assuming that there is no absolute maximum temperature? But you do probably know more than me so I'm not the one to talk here. – Insane Jan 12 '15 at 0:03
• Insane, yes, your question is posed slightly differently, but the answers will be the same. See, for example, this answer to the question I linked for an answer that is a literal answer to your question here. – ACuriousMind Jan 12 '15 at 0:05
• If you like this question, you may also enjoy reading this Phys.SE post. – Qmechanic Jan 12 '15 at 1:39

There is a "high temperature" limit, it is actually the same point. 5K is hotter than 1K, and 1000K is hotter than both. The temperature $\pm \infty$ is actually the same temperature, and for negative temperatures you get hotter the closer you get to zero (from below).

Most systems cannot reach negative temperatures, but even if you can you can't get all the way to zero from either direction.

If this seems weird, note that $$\frac{1}{kT}=\frac{d S}{d E},$$

So if giving a little bit of energy $E$ gives a large increase in entropy $S$, then $1/kT$ is large (and positive), so $T$ is small (and positive). If a little bit of energy gives no change in entropy (not possible for most systems) then $1/kT$is zero, so $T=\pm\infty$. If a little bit of energy gives a large decrease in entropy $S$, then $1/kT$ is large (and negative), so $T$ is small (and negative).

These changes in entropy can never be infinite because entropy is a real and finite thing and we don't make vanishingly small changes in energy, so there is no infinite rate $\frac{d S}{d E}$. And that's why temperature is never zero.

edit to discuss when negative temperatures can occur

All you need for negative energy is that the range of small-and-practically-accessible energy exchanges can only cause the entropy to get smaller when it accepts energy. This makes it statistically favorable to give energy to positive temperature things (then both entropies can go up), so it is hotter than all of them (so negative temperature is hotter than positive temperature). This might not be because the negative temperature system has components with a strict maximum energy, just that those higher entropy states are not accessible. For instance if your collection of two state systems don't have enough kinetic energy to combine to create electron-positron pairs it doesn't matter that with a large amount of additional energy they could increase their entropy by doing so, as long as that isn't going to happen. Statistical physics is about what will happen on time scales long enough for quasi-equilibrium to occur.

• +1, but I'm a bit wary of your last two sentences. It is conceivable, though highly unwonted, that a system in its time evolution could suddenly access several degenerate (equal energy) states that it couldn't beforehand. If so, it could spontaneously shuffle its energy around between these degenerate states, thus leading to an increase in entropy whilst its internal energy stayed precisely constant ($\mathrm{d}E=0$). This would correspond to an exactly zero temperature. IMO "Temperature is never zero" is more accurately said "a system's temperature can never reach nought through ..." – Selene Routley Jan 12 '15 at 0:42
• ... the use of a heat pump dumping the system's heat to a nonzero temperature reservoir". But there is no physical reason why a system can't be at a zero temperature, there's just no way to make it reach this state when the system is surrounded by a nonzero temperature universe. See my answer here for a more quantitative discussion. – Selene Routley Jan 12 '15 at 0:45
• Also, it might be worth pointing out that infinite and negative temperatures can only be reached for systems whose constituents have strict upper limits to their energy states. An ensemble of quantum harmonic oscillators cannot reach infinite or negative temperature, whereas an ensemble of two-state quantum systems can do both these things: see my answer here – Selene Routley Jan 12 '15 at 0:49