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  How is temperature defined for interacting systems?

+ 1 like - 0 dislike
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Temperature is easy to define for closed systems. By definition, for closed systems (or systems with negligibly small interaction with the environment) temperature is (see Chandler, for example) $$ \frac{1}{T}\equiv\left(\frac{\partial S}{\partial U}\right)_{V,N},\tag{1} $$ where $V$ is the volume, $N$ is the number of particles, $U$ is the energy and $S$ is the entropy of the system.

One can prove that two noninteracting (or rather weakly interacting) systems at thermodynamic equilibrium have the same temperature using the principle of maximization of entropy. The principle says that when the total energy $U$, number of particles $N$ and volume $V$ are fixed for the composite system, the total entropy is maximized at equilibrium. The proof of equal temperatures is very simple. At equilibrium the entropy is at maximum, and therefore, if we transfer small amount of energy $\delta U$ from one system to another, the total change in entropy is going to be zero: $$ \delta S=\frac{1}{T_1}\delta U_1+\frac{1}{T_2}\delta U_2=\left(-\frac{1}{T_1}+\frac{1}{T_2}\right)\delta U, $$ which is true only for $T_1=T_2$.

My question, for interacting systems, where the total energy is expressed as $$ U=U_1+U_2+U_{12},\tag{3} $$ where $U_{12}=f(U_1,U_2)$ is the interaction energy, is there an accepted definition for temperature?

Note. The temperature cannot be defined as $$ \frac{1}{T_1}=\left(\frac{\partial S_{tot}}{\partial U_1}\right)_{V,N},\tag{4} $$ with $S_{tot}=S_1+S_2+S_{int}$, because with such definition, since $\delta U_1 +\delta U_2=-\delta U_{int}\neq 0$, the temperatures of both systems are not going to be equal at maximum entropy.

Personal idea. I am not sure if this way is a proper one, but here is my idea. We could define temperature through a tunable turn-on-off of the interaction. We could imagine that interaction between two systems is governed by a parameter $\lambda$. The value $\lambda=0$ corresponds to the no interaction case, and $\lambda=1$ corresponds to the full interaction. The interaction energy would be then a function of $\lambda$ (probably a monotonous one): $$ U_{int}=f(U_1,U_2,\lambda). $$ This relationship can be inverted: $$ \lambda=F(U_1,U_2,U_{int}).\tag{5} $$

The total entropy is a function of $U_1$, $U_2$ and $\lambda$ $$ S_{tot}=G(U_1,U_2,\lambda). $$ Expressing $\lambda$ through $U_{int}$ from Eq. (5), we obtain that $S_{tot}$ is a function of $U_1$, $U_2$ and $U_{int}$: $$ S_{tot}=S(U_1,U_2,U_{int}). $$ The temperature would then be defined as $$ \frac{1}{T_1}=\frac{\partial S}{\partial U_1}.\tag{6} $$ With such definition, the variation of $S_{tot}$, $$ \delta S_{tot}=\frac{1}{T_1}\delta U_1+\frac{1}{T_{2}}\delta U_2+\frac{1}{T_{int}}\delta U_{int} $$ is zero for the fixed total energy if $T_1=T_2=T_{tot}$ (this is not the only solution though). In other words, with the above definition of temperature, the equal temperature of different parts of the system is consistent with maximum entropy at equilibrium.

Does anyone know if anyone introduced temperature for interacting systems in the literature?

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Pavlo. B.
asked May 10, 2021 in Theoretical Physics by Pavlo. B. (5 points) [ no revision ]
retagged 2 days ago
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@Pavlo.B. Note that if you do not assume that $U_{1,2} \ll U_1 + U_2$, then the energy fails to be extensive, a basic assumption of thermodynamics. From the point of view of Statistical Mechanics, if the interaction between the constituents of both systems decays fast enough with the distance, then one can show that $U_{1,2}$ is indeed negligible for macroscopic systems.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Yvan Velenik
@BioPhysicist I used that comment partly to "Leave constructive criticism that guides the author in improving the post", and partly to "Add relevant but minor or transient information". I don't think it qualifies as an answer.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
@pglpm I guess it's not obvious to me what improvement you were wanting the OP to make explicitly.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user BioPhysicist
@YvanVelenik The question was aimed at the case when $U_{12}$ is not negligibly small. I understand that extensivity fails. But non-extensivity does non prevent one from deriving pressure difference due to surface curvature+surface tension from thermodynamic reasons (in thermodynamic limit such difference does not exist). I am looking for some similar corrections to the temperature definition, like the effect of the pressure difference between small droplets and gas due to surface tension

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Pavlo. B.
I see. I am familiar with the corresponding problem with pressure (Gibbs-Thomson formula and related results), but have never come across similar questions for the temperature.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Yvan Velenik
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@YvanVelenik Please only use the comments to ask for clarifications or suggest edits to the post. Otherwise, if you think you have something useful for the OP you want to say you should put what you have to say in an answer.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user BioPhysicist
@pglpm Please only use the comments to ask for clarifications or suggest edits to the post. Otherwise, if you think you have something useful for the OP you want to say you should put what you have to say in an answer.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user BioPhysicist

1 Answer

+ 1 like - 0 dislike

Your question seems to have the hidden assumption that temperature is defined as the partial derivative of entropy with respect to energy, but that is not a universal point of view in general thermodynamic applications. Here I summarize two possible approaches to your question: from an equilibrium, generalized Gibbsian viewpoint, and from the viewpoint of non-equilibrium continuum thermomechanics. References are listed at the end.

Generalized Gibbsian thermostatics

In the kind of equilibrium situation that you seem to consider, the generalized "Gibbsian" point of view is that

  1. We state which quantities define the thermodynamic state (and therefore the system), in a non-redundant way. In your case it seems you can take any three of the energies $U,U_1,U_2,U_{12}$, the remaining one being determined them, plus any other relevant thermodynamic quantities. Of course we can change the set of basic quantities to equivalent ones; this corresponds to choosing a different coordinate system on the manifold of thermodynamic states.

  2. We give the "fundamental equation" (Gibbs's and Weightman's terminology) that specifies how the entropy of the system depends on the thermodynamic variables. This function is usually required to be convex, but this property can be problematic or undefined in non-extensive systems, and some authors seem to drop this requirement to deal with phase transitions, for example (see eg Wightman, Landsberg & Tranah).

Then any equilibrium state is determined by maximizing the entropy function with respect to the constraints imposed on the system, expressed as functional relations between the thermodynamic variables. Of course it can happen that there's no maximum or that there are many maxima (this can also happen in standard textbook cases, see eg Callen, Appendix C p. 321). This is a general point of view that does not require extensivity or similar properties. With extensive systems additional properties of the entropy function are usually required.

From this point of view temperature becomes a somewhat secondary concept, and you could define auxiliary temperatures, if they are useful, in a "$1/T=\partial S/\partial U$" way. In your case you could define three inverse temperatures $\beta_1 = \partial S/\partial U_1$, $\beta_2 = \partial S/\partial U_2$, $\beta_{12} = \partial S/\partial U_{12}$. If your system has a constrained total energy but the energies $U_1$, $U_2$, $U_{12}$ are unconstrained, then maximization of entropy on the constrain submanifold of the state space leads to $\beta_1=\beta_2=\beta_{12}$ in the usual way. However, note that this $\mathrm{d}S\rvert_{\text{constr. submanifold}} = 0$ point might not be a maximum, and thus not an equilibrium point. See the discussion in Lynden-bell, especially at the bottom of p. 295.

There's also the tricky point to say which quantities should be kept constant in these partial derivatives: usually they are extensive quantities, but your present system is presumably non-extensive. There are studies of equilibrium non-extensive systems (typically gravitational ones), see for example Landsberg & Tranah and Lynden-Bell; but I can't find how they approached the question of the partial derivatives. And I don't know how these studies have developed in recent times.

References for this approach:


Continuum thermomechanics

This is a theory of time-dependent, space-dependent, non-equilibrium thermodynamical and mechanical processes. It's a field theory: physical quantities depend on space and time. Equilibrium thermostatics is obtained in the special case where quantities are time-independent and have a uniform spatial distribution. Great introductions to this theory are Astarita's and Müller & Müller's books.

Temperature here is taken as a primitive quantity which depends on space and time, so there's no question of "defining" it in terms of other quantities. In particular, temperature is a property of matter at every point in space, not of a "subsystem". It can have discontinuities, however: for example we can have $T=T_1$ in a region of space, $T=T_2$ in an adjacent region, and $T=T_{12}$ on the surface separating the two regions. In the case of mixtures of different chemical species it would be possible to define a (spatially dependent) temperature for each of them, but such a generality is usually not pursued; see eg Samohýl & Pekař chap. 4, or Truesdell Lecture 5 and appendices to it.

The "$1/T=\partial S/\partial U$" relation is not generally valid a priori in this theory; see for example Astarita chap. 2 and Samohýl & Pekař chap. 2.

Continuum thermodynamics typically deals with situations with short-range forces, so energy is extensive. Interaction energies may be considered at surfaces (surface tension). But, again, this does not interfere with how temperature is used. See for example Astarita chap. 6.

There is a generalization of this theory that deals with energetically non-extensive systems – typically gravitational ones. Energy and entropy have a "doubly spatial" dependence in this approach. But temperature remains a quantity associated with each single spatial point. See the studies by Gurtin & Williams; unfortunately I'm not updated about recent developments of these studies.

References for this approach:


Final comments

I personally prefer the second approach, because it covers more general situations. And also because in my opinion the "$1/T=\partial S/\partial U$" definition of temperature is vacuous, since it defines temperature in terms of non-directly measurable quantities. Actually we need temperature as a primitive in order to indirectly measure energy: see for example the discussion in Müller & Müller § 2.3. So the "definition" above involves some circularity.

Very insightful further references about all these questions:

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
answered May 12, 2021 by pglpm (590 points) [ no revision ]
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@Pavlo.B. "one usually tries to introduce temperature precisely in a way to make temperature equal everywhere at equilibrium" is a strange statement. Historically temperature wasn't introduced that way. And if you pay attention to how you use temperature in your everyday you'll notice that you don't use it that way either. The books by Astarita and Chang above are very beautiful to read and open many horizons. Some textbooks in thermostatics limit themselves to very abstract and oversimplified systems (gas in a box) to introduce basic concepts, but unfortunately never rise above that level.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
@Pavlo.B. You can have an adiabatic separation within the parts of a closed system, and those parts can be at different temperatures, and the system is in equilibrium. This is not an uncommon situation. "Equilibrium" always needs a specification of which constraints operate on the system. An ideal "$(U,V,N)$" gas without any constraints whatsoever has an undetermined state, because any combination of $U\to\infty,V\to\infty,N\to\infty$ limits will make the entropy increase. Now, if you say for example "but $N$ is fixed" – well that's a constraint.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
"You can have an adiabatic separation within the parts of a closed system, and those parts can be at different temperatures". Of course, if you do not allow parts of the system to exchange energy, temperature does not have to be equal. All derivations of equal temperatures (that I know) rely on the free exchange of energy. I thought you meant "even if we allow parts of the system to freely exchange energy, the temperature needs not to be equal". So, if you allow a free energy exchange, then the temperature has to be equal, right? (just want to make sure I understand you correctly)

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Pavlo. B.
@Pavlo.B. I've always been curious about the possibility you mention, and personally I don't rule it out in the case of large gravitating systems (there were some works by Müller that possibly studied that possibility). But best to avoid longer discussions in comments: I warmly recommend you check the references given above, I'm sure you'll find them interesting and much more illuminating than my comments!

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
@Pavlo.B. I've expanded and structured the answer and added a couple of references, I hope they'll be useful.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user pglpm
Most recent comments show all comments
I like the definition of temperature through the the partial derivative over the constraint functions $U_1$, $U_2$ and $U_{12}$. I am not sure how I feel about that Gurtin & Williams paper. It is pretty hard to get through, but they do not seem to define temperature in their consideration and just treat it as a separate field. In other words, temperature, entropy and energy seem be able to vary independently in their consideration.

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Pavlo. B.
They also introduced this interesting "specific binding entropy", where they they break the entropy into pairwise interaction entropy. Do you know if the entropy can be expanded like that?

This post imported from StackExchange Physics at 2025-01-22 10:48 (UTC), posted by SE-user Pavlo. B.

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