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When I studied Thermodynamics the way I learned things and the way I understood them is that temperature is a quantity that appears and is relevant just when studying macroscopic matter.

Indeed, all the framework I've learned to deal with temperature was connected to this "macroscopic matter approach", so that it all starts with a fundamental relation $U = U(S,X_1,\dots,X_n)$ where $S$ is the entropy and $X_i$ are extensive parameters of the macroscopic system.

In that setting the temperature is:

$$T = \dfrac{\partial U}{\partial S}.$$

So there are some remarks here:

  1. $T$ is a function of the parameters describing the macroscopic body;
  2. $T$ is not a function defined pointwise in space;
  3. $T$ only makes sense for a very specific macroscopic body, namely, the one described by the extensive parameters $X_1,\dots,X_n$ and whose fundamental relation is $U = U(S,X_1,\dots,X_n)$.

Even when I studied statistical mechanics everything remained the same, the only difference was that statistical mechanics gave a way to compute $S$ or the other potentials, starting from microscopic considerations.

Now, I've seem some times now some discussions where temperature is involved in phenomena related to fields and particles, like when talking about the big bang and the evolution of the universe after it.

But in this framework I learned it simply doesn't make any sense bringing these things together.

So, how can temperature ever relate to phenomena involving fields and particles, and how do we deal mathematically with temperature in these situations, since, as I said, the traditional framework, just relates temperature to equilibrium states of macroscopic matter?

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  • $\begingroup$ I'm not so sure what your exact question is. The notion of the thermal state $\exp(-\beta H)$ carries over exactly to quantum field theory - "thermal" quantum field theory is QFT for that state, not for the vacuum state, and even more generally the Keldysh-Schwinger formalism deals even with non-equilibrium situations. This is what one directly finds when trying to search for "thermal field theory". Is your issue that the notion of temperature in those theories seems a bit more general than what you know so far? $\endgroup$ – ACuriousMind Aug 21 '16 at 21:50
  • $\begingroup$ Yes, I believe that's the issue. As I said, the way I learned about temperature is as I wrote, the definition $T = \partial U/\partial S$. This seems pretty limited, because even the simple example of a scalar field $\phi$ representing temperature used everywhere doesn't make sense on this framework. So yes, I believe the issue is that the notion of temperature used in theories like that is more general than what I've seem. $\endgroup$ – user1620696 Aug 21 '16 at 21:54
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From the statistical mechanics point of view, the temperature $T$ is just a parameter entering the definition of the equilibrium state $$\rho = \frac{{\rm e}^{-\beta H}}{{\rm Tr}\left[{\rm e}^{-\beta H}\right]},$$ where $\beta = 1/k_B T$ and $H$ is the Hamiltonian of the system in question. This microscopic definition agrees with the classical thermodynamics definitions if you identify the mean values of observables like the energy $U = {\rm Tr}\left[\rho H \right]$ with the extensive thermodynamic parameters. However, the statistical mechanics approach is more general, because there is no requirement for the system to be macroscopic: it could be a single particle or a quantum field etc.

However, the state of the system must be of the form above (or, more generally, the equivalent Gibbs form from another ensemble, e.g. grand canonical). In other words, the entropy of the system must be maximised subject to the constraint of fixed $U$ (plus any other extensive conserved quantities e.g. particle number), which uniquely defines $\rho$ and $\beta$ (plus any other thermodynamic potentials e.g. chemical potential). Any time someone is discussing thermodynamic temperature in any context, they are implicitly assuming that the system is an equilibrium (maximum entropy) state. Otherwise $T$ is undefined.

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  • $\begingroup$ Thanks for the answer @MarkMitchison. Where can I find a discussion of this point of view on temperature? Is there some book or article which shows how this works and how can we introduce temperature in this way? When I studied Statistical Mechanics, temperature was a concept carried over from Thermodynamics, defined just as $T = \partial U/\partial S$. I belive this approach, of introducing temperature as this parameter might make more sense indeed. I just couldn't find any resource deepening this exposition. Thanks again for the aid. $\endgroup$ – user1620696 Sep 6 '16 at 19:24
  • $\begingroup$ @user1620696 I am unsure about references, but actually I'm not introducing anything new here. The temperature is defined in terms of the derivative of energy $U$ w.r.t. entropy $S$ as you write in your post. In equilibrium, this is the same as what I wrote (and the temperature is only defined in equilibrium). My point here is that, in general, one must interpret thermodynamic variables like $U$ and $S$ as averages over a microscopic probability distribution. $\endgroup$ – Mark Mitchison Sep 7 '16 at 16:12
  • $\begingroup$ Macroscopic thermodynamics emerges in the limit where the system is so large that any measurement is practically equal to the average. But that does not mean that these concepts cannot also be applied to microscopic systems, where there are significant fluctuations away from the average. The only difference is that a small number of thermodynamic (average) variables no longer suffices for a complete description of a microscopic system: instead, you need the full probability distribution (i.e. the quantum state). $\endgroup$ – Mark Mitchison Sep 7 '16 at 16:14

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