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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.

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    $\begingroup$ What does "thermal equilibrium state" mean, if not that they are at the same temperature? $\endgroup$ Commented Apr 9, 2023 at 17:59
  • $\begingroup$ Conventional thermodynamics is axiomatic. The three laws are axioms in a mathematical sense. $\endgroup$ Commented Apr 9, 2023 at 18:07
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    $\begingroup$ The phrasing sounds strange. Maybe the author assumes membranes are like pistons free to move, implying the same pressure for each gas. $\endgroup$
    – Benoit
    Commented Apr 9, 2023 at 22:08
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    $\begingroup$ 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. $\endgroup$ Commented Apr 10, 2023 at 14:57
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    $\begingroup$ Related: physics.stackexchange.com/q/388318/2451 $\endgroup$
    – Qmechanic
    Commented Apr 10, 2023 at 15:25

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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).

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