# Can temperature be a complex number?

Is it possible for a temperature to be a complex number? I want to say "no" but I can't be so sure. If it is possible I would like to know of an example. I found an interesting article which treats the Riemann Zeta function as a partition of a statistical-mechanical system called a Riemann gas. I don't know precisely this is related, but I wasn't aware that the Riemann Zeta function and Its connection to primes had physical significance.

Edit: If anyone could post an example of an explicit calculation where we get a temperature corresponding to some number in $\mathbb{C}$, I would be very interested in the result

• Sure. How does heat flow from 300iK to 280iK? What happens between (300+20i)K and (300-20i)K? Jun 9, 2015 at 22:06
• @StVincent the question is can a thermodynamic system exist that has complex eigenvalues? I can think of one example - the Rijke tube. Temperature can oscillate as a standing wave in that system - so I think the answer is yes. Jun 9, 2015 at 22:18
• I am simply asking how you define temperature. The second law defines it as the force that makes heat flow, which makes it automatically a partially ordered relation. So I am asking how an imaginary temperature makes heat flow, given that complex numbers are not naively partially ordered. Jun 9, 2015 at 22:21
• @VladimirKalitvianski: Temperature is what makes heat flow. Period. No human imagination necessary. Unless, of course, you are among those who imagine that hot bodies in contact with cold bodies will not cool down unless the imagination of Vladimir Kalitvianski is involved? In that case I would ask you to cool my coffee down for me, please, it's too hot. Jun 9, 2015 at 22:43
• @CuriousOne: It is the gradient of $T$, not temperature itself who cools down your coffee. And I speak of importance of temperature fluctuations existing in the Nature which may puzzle you if you forget them. Jun 9, 2015 at 22:49

In some sense yes. The temperature is defined as an imaginary time in Matsubara Green's functions or some path integrals. Thus, a negative inverse imaginary temperature can be considered as a time. Here is a quotation from Alexander Altland, Ben Simons "Condensed Matter Field Theory":

"Thus, real time dynamics and quantum statistical mechanics can be treated on the same footing, provided that we allow for the appearance of imaginary times."

Edit: If anyone could post an example of an explicit calculation where we get a temperature corresponding to some number in C, I would be very interested in the result

The calculations might be for a quantum statistical system being out of thermal equilibrium in order to get both real (imaginary) and imaginary (real) time (temperature) in same equation.

I am not particularly familiar with the primon gas you are linking to, but similar ideas have been tossed around for a long time; see, for example this page for many references (including the topic you mention). The first two topics (quantum mechanics and statistical mechanics) are particularly relevant to your question; I'll concentrate on the second one, with which I am more familiar.

One of the early motivations was the Lee-Yang circle theorem. The latter states that the partition function of the Ising model, seen as a function of a complex magnetic field $h$, has all its zeros on the imaginary axis. The result extends to a larger class of models, and it was natural to wonder whether the Riemann hypothesis could be recast into this language for a suitable model.

Now, on the more general issue of complex temperatures (or magnetic field, etc., for that matter). As mentioned above, considering a complex magnetic field plays a crucial role in the Lee-Yang approach to phase transitions. This should not be surprising: one manifestation of phase transitions is as non-analyticities of thermodynamic potentials. In order to assess analyticity of the latter, one has to consider them as functions of complex parameters. Such a point of view does not only provide one of the general approaches to phase transitions (a phase transition occurs when zeros of the partition function accumulate near the real axis in the thermodynamic limit), but also provides a wealth of information about the system.

Note that, if Lee and Yang have considered complex magnetic fields, it is also natural to look for zeros of the partition function as a function of a complex temperature. This was done by Fisher in the 1960s (google "Fisher zeros").

To conclude, I would be very surprised if the Riemann hypothesis turned out to have deep implications in physics. However, it is conceivable that analogies between physical problems and the Riemann hypothesis might shed light on the latter.

• Thus, the complex temperature is a mathematical trick, and not a quantity that has a physical sence? Nov 5, 2021 at 11:24
• Yes, that's right. Well, there are (very indirect) links to experimentally observable phenomena: see this paper or this one for example. But, yes, this is a useful mathematical "trick", but a very natural one, given that phase transitions are closely related to the analytic properties of thermodynamic functions, and the latter are best investigated and understood in the complex plane. Nov 5, 2021 at 12:31