Postulate of macroscopic thermodynamics and its uniqueness

I was learning statistical mechanics and referred to the book "statistical thermodynamics" by Tien, C. L.; Lienhard, J. H.

With reference to Herbert Callen's book, Postulates of Macroscopic thermodynamics was given.

Callen has combined the physical information that is included in the Laws of thermodynamic into a set of four postulates.

The first Postulate is:

There exist certain states (called equilibrium states) of simple systems that macroscopically characterized completely by Internal Energy ($$U$$), Volume($$V$$), and number of moles ($$N_1,N_2,\cdots N_r$$) of the $$r$$ chemical components.

In the context of this postulate, it has been mentioned that

Suppose that an isolated system is made up of several subsystems which may or may not be isolated from one another. The problem is then that of determining the final state of this composite system when any such internal constraints as might exist (e.g. walls of any kind) are removed. The first postulate implies that such a final state will be unique but gives no way of determining it.

Why do we need these postulates and how did these postulates make our life easy in statistical thermodynamics. I think these postulates a slight rearrangement of laws of thermodynamics?

I didn't Understand How this first postulate implies that the final state of the system will be unique. and in the first place what do we mean by unique here. I think the most probable macrostate will be the final state, then from where the question of uniqueness comes?

• Sometimes it is useful to have more than one way to state basic ideas in thermodynamics. I would say the different statements do not replace one another but they may illuminate one another. Mar 14 '21 at 11:35

Callen's postulates are intended to summarize the laws of thermodynamic into a set of statements reasonably equivalent to the classic principles and much closer to the needs of theoretical analysis. They are definitely a better starting point to establish a link between Thermodynamics and Statistical Mechanics. Actually, in the preface to the second edition of his textbook, Callen wrote:

... formulating thermodynamics so that its macroscopic postulates are precisely and clearly the theorems of statistical mechanics...

What about the uniqueness of the final equilibrium state? The first thing to avoid is to mix the statistical mechanics description and the thermodynamic description. Their consistency (and the conditions for their consistency) has to be justified only after the two separate points of view have been separately understood. Mixing the two approaches would make it difficult to clearly see which results depend on the microscopic description and macroscopic one. Of course, the two descriptions do have a contact, but it is important to understand that its contact is not a full overlap. The description of many-body systems provided by statistical mechanics is not always coinciding with the description of classical thermodynamics.

Therefore, from the perspective of classical thermodynamics, the statement about the uniqueness of the equilibrium state must be intended as equivalent to saying that the equilibrium state of a simple system is a function of the quantities $$U$$, $$V$$, and $$N$$. Here, function of should be intended in its mathematical meaning: for each triple of values $$(U, V, N)$$, it exists one and only one equilibrium state. In principle, there would be no reason for excluding the possibility that two or more equilibrium states could exist in correspondence with the same triple of values. It is a fact from experience that this is not true for simple systems. If, in a real experiment, one would find more than an equilibrium state associated with the same values of internal energy, volume, and number of moles, the only conclusion would be that that system is not a simple system, and additional variables are required to characterize that equilibrium state.

Why do we need these postulates and how did these postulates make our life easy in statistical thermodynamics.

This is a sort of essay question which is asking the importance of statistical mechanics. And I won't answer it because the importance of the subject can only be understood when you use it! So to get the answer you just need to read the book (any textbook). You will easily find that thermodynamics fails to describe black-body radiation or solids etc.

I think these postulates a slight rearrangement of laws of thermodynamics?

No! This is not true. The fundamental law of statistical mechanics says that

In a state of thermal equilibrium, All accessible microstates of the system are equally probable.

which is external input to these mechanics and can't be proved.

I didn't Understand How this first postulate implies that the final state of the system will be unique.....

You can understand it by example, Suppose a system of spins (something like Ising model) where each spin can point up or down. You can divide them into sub-systems (initially isolated from each other) and suppose at any instance, you make them interact with each other. The system will evolve under some Hamiltonian. Then the first principle says that after some time this system will come to an equilibrium state, which has a unique parameter $$U,\cdots$$. But this first principle unable to tell you what these values will be!

The uniqueness means the unique value of the thermodynamic parameters at the state of equilibrium.