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I have been working on an axiomatic approach to thermodynamics, and tried to follow the footsteps of Theodoro Frankel using his little book, The Geometry of Physics.

The passage appears before introducing the first law of thermodynamics. In case you don't have a copy on hand, I put an extract below:

Consider, for example, a system of regions of fluids separated by "diathermous" membranes: membranes that allow only on the passage of heat, not fluids. We assume the system to be connected. We assume that each state of the system is a thermal equilibrium state. Let $p_i, v_i$ be the (uniform) pressure and volume of the $i^\text{th}$ region. The "equations of state" (e.g. $p_i v_i = n_i R T_i$) at thermal equilibrium will allow us to eliminate all but one pressure, say $p_1$; thus a state, instead of being described by $p_1,v_1,...,p_n v_n$, can be described by the $(n+1)$-tuple $(p_1,v_1,v_2,...,v_n)$.

As far as I understood, the fluid as a system is partitioned into $n$ connected components that each is a thermal equilibrium state. Therefore, there are $n$ different equilibrium states.

My question is that how can the number of variables be reduced from $2n$ to $n+1$ by introducing equations of state's? These states are essentially at different temperatures. I would feel comfortable if they were at the same temperature but it seems not.

P.S. I have noticed another post, Principle of Caratheodory and The Second Law of Thermodynamics, but it seems not close to my question.

Qmechanic
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Kevin Kwok
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    What does "thermal equilibrium state" mean, if not that they are at the same temperature? – Marius Ladegård Meyer Apr 09 '23 at 17:59
  • Conventional thermodynamics is axiomatic. The three laws are axioms in a mathematical sense. – FlatterMann Apr 09 '23 at 18:07
  • Truesdell: Rational Thermodynamics. You will not a find single complimentary sentence about the Caratheodory style mathematization of thermodynamics; Truesdell's is axiomatic in a strict mathematical sense. And while @FlatterMann is right to say that the three laws are axioms and in that sense thermostatics (not dynamics) is axiomatic in all its several versions (Kelvin, Clausius, Caratheodory, Gibbs, etc.) there is a fair amount of handwaving involved with all of them that is rightly criticized by Truesdell and his followers. – hyportnex Apr 09 '23 at 18:40
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    The phrasing sounds strange. Maybe the author assumes membranes are like pistons free to move, implying the same pressure for each gas. – Benoit Apr 09 '23 at 22:08
  • @hyportnex I agree that thermostatics (or quasi-thermostatics) is a much better term for what TD really is. I also have to agree that the conventional presentation waffles a lot. What I don't know is whether the problems with TD are truly mathematical at the core. Given the need to treat even the most simple heat flow problem (black body radiation) quantum mechanically, it's not obvious that "better axioms" can resolve them. I am somewhat skeptical. I will try to read some Truesdell, then. – FlatterMann Apr 09 '23 at 22:29
  • @FlatterMann Truesdell is an amazingly entertaining writer. Try to get hold of the 1984 2nd edition, his "The Tragicomical History of Thermodynamics" is also truly marvelous. I must warn you though that not everybody is as enthused with his style as I am, I sure would not want to have gotten in a public argument with him. He was like an Olympic saber champion in a debate, in private he was the kindest gentleman, though. His main concern was creating a thermodynamical theory that would be a real match for mechanics with the mathematical depth in the Euler-Lagrange style. – hyportnex Apr 09 '23 at 22:44
  • @hypertonex And there is the problem right away: Euler-Lagrange is already a subset of mechanics, and so is Hamilton. They are excellent methods for conservative force problems. Those, however, are not the only possible problems in mechanics. TD falls apart at similar kinds of borders of the theory. One can certainly axiomatize a very limited number of physical scenarios, but I am not sure the program carries much farther than conventional TD does. I will give him a chance, though. – FlatterMann Apr 09 '23 at 23:18
  • @Benoit I think you may be write, at least this can explain the situation. $p_1=p_2$, $p_2=p_3$, ..., $p_{n-1}=p_n$ in total $(n-1)$ constraints. – Kevin Kwok Apr 10 '23 at 02:25
  • @MariusLadegårdMeyer Yes. But it is the $n$ subsystems of the system separately in thermal equilibrium, so there are $T_1,...,T_n$ temperatures. – Kevin Kwok Apr 10 '23 at 02:28
  • @hyportnex you mentioned that, the three laws are axioms and in that sense thermostatics (not dynamics) is axiomatic, how about the remaining one? By the way, I have just got a copy of Truesdell and will read it right away. – Kevin Kwok Apr 10 '23 at 02:35
  • What does it mean for a single subsystem nr. $n$ to be in thermal equilibrium? What is it in equilibrium with? A bath outside the $n$ subsystems? I find it more plausible the author means with the other subsystems. – Marius Ladegård Meyer Apr 10 '23 at 04:26
  • @MariusLadegårdMeyer I was thinking of the system as a whole is a fluid, and we partition it in $n$ differently regions, or subsystem, labeled by $i$. Each subsystem consists of many molecules in the order of a mole or more. By each (of states) is in thermal equilibrium, I think it means that the subsystem itself is in thermal equilibrium in the sense that it can be described solely using some intensive parameters such as pressure $p_i$ and temperature $T_i$. – Kevin Kwok Apr 10 '23 at 05:09
  • Thermodynamic equilibrium means equal temperature. That's the second law: heat only flows from hot to cold (without anything else happening). More precisely: heat will flow from hot to cold without anything else happening until there is no more hot and cold. If you are proposing to use an independent definition of temperature for each phase, then it's not clear what "temperature" is even supposed to be and the second law goes out the window. – FlatterMann Apr 10 '23 at 07:55
  • @FlatterMann I am not sure whether I got it right. It means that once the author said "We assume that each state of the system is a thermal equilibrium state." It implies that the whole fluid is at the same temperature. Otherwise it doesn't make sense? I was thinking about the system is still in a transient state so that it is as a whole non-equilibrium, but there are different equilibria in different regions. – Kevin Kwok Apr 10 '23 at 09:06
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    TD says very little about non-equilibrium states. What you are trying to do is to find a recipe for non-equilibrium TD. That is going to be impossible because there isn't one. – FlatterMann Apr 10 '23 at 14:57
  • @FlatterMann what I meant by "Euler-Lagrange" is the style not the substance, in both style and substance it is more akin to D'Alembert's principle (this is my view) which, according to Lanczos, is the most general formulation of mechanics. Truesdell's program from the 50's on was to bring differential equations, tensors, and other analytical tools in, as are ubiquitous in the mechanics of deformable bodies, and leave out concepts such as "engine", "no other effect than", etc., out along with the unorthodox use of a variety of non-equivalent infinitesimal quantities. – hyportnex Apr 10 '23 at 15:15
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    Related: https://physics.stackexchange.com/q/388318/2451 – Qmechanic Apr 10 '23 at 15:25
  • The three laws of thermostatics I was referring to: the $0^{th}$ law of thermal equilibrium being transitive; the $1^{st}$ law of energy conservation; the $2^{nd}$ law of the entropy being a quasi-conserved, nondecreasing quantity. What is $4^{th}$? Truesdell working with Noll, Coleman, Gurtin, Ericksen, etc., created a general mathematical framework they called "rational thermodynamics" to go beyond thermostatics, that is beyond equilibrium processes. It is not a popular movement and has many critics among this review of prevalent methods is Lavenda:"Thermodynamics of irreversible processes" – hyportnex Apr 10 '23 at 15:27
  • @hyportnex My comments simply reflect the reality that there are no perfectly self-consistent theories that cover a wide area of macroscopic physics. Every macroscopic theory inevitably runs into the boundary to microscopic physics and its emergent effects. Mechanics runs into friction. TD runs into convection, turbulent flow and chaos. No amount of mathematical reformulation can help with that because reality is "just isn't so". Truesdell was a decade or two before the non-linear physics revolution. Haken would have told him, with physical examples, that he was wasting his time. – FlatterMann Apr 10 '23 at 16:26

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Out of context, it is unclear what the author means. Maybe he is trying to prove that a global equilibrium implies that all temperatures are the same, but he does not assume it from the start. The text really suggest that all fluids are in thermal equilibrium with each-other.

You may be interested in this text: https://arxiv.org/abs/cond-mat/9708200

It is an axiomatic approach that deals with systems from a pure macroscopic approach, not assuming matter is made of molecules and without statistical mechanics. I understand it as a modern continuation of Caratheodory’s ideas. (I have only read bits of it, not everything is easy to read).

Benoit
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