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For a single-component system, why are the energy, volume, and number of particles sufficient for describing the thermodynamics of the system? Why just three variables and those three variables in particular?

In the book that I am using (Callen, $\textit{Thermodynamics and an Introduction to Thermostatistics}$) he postulates that the macroscopic equilibrium state is characterized by the energy, volume, and particle numbers of its components, but what is the reason for this?

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Typically, the state variables used to identify a macrostate are N, V, E for an isolated system, or you can replace E with T if the system is in contact with an heat bath. Other choices are also possible (the different formulations are related by the Legendre transform). Note that other substances may require an enlarged set of state variables (for example, magnets or superfluids, see https://arxiv.org/abs/cond-mat/0405111 ). Generally, apart from V (that sets the "size") and the internal energy E (that is related to the microscopic Hamiltonian), the other variables should be "conserved quantities" under the (possibly dissipative) dynamics of the system: for a simple system N is a conserved quantity (it is a "Noether charge"), other systems (like superfluids) may have extra conserved quantities.

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    $\begingroup$ In the case of an isolated system, is the choice of N, V, and E known only from experiment? $\endgroup$
    – user2561523
    Feb 1, 2021 at 2:27
  • $\begingroup$ It's more a theoretical thing. If the system is isolated it does not exchange energy, so E is well defined (but in experiments E is hardly accessible). Also N is not precisely known (it is of the order of the Avogadro number). Things are introduced theoretically in a certain way (say with E, N, V) and then we use the Legendre transform to change variables (experimentally typically you know V, T, n where n is the density or P, T, n). In the last case of PTn you do not know how "big" is the system but this set of variables is useful to do hydrodynamics. $\endgroup$
    – Quillo
    Feb 2, 2021 at 14:36
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A flippant, but not entirely inaccurate, answer is that those three variables were chosen because it was convenient.

From the Wikipedia page for Thermodynamic State:

A thermodynamic system is a macroscopic object, the microscopic details of which are not explicitly considered in its thermodynamic description. The number of state variables required to specify the thermodynamic state depends on the system, and is not always known in advance of experiment; it is usually found from experimental evidence. The number is always two or more; usually it is not more than some dozen. Though the number of state variables is fixed by experiment, there remains choice of which of them to use for a particular convenient description; a given thermodynamic system may be alternatively identified by several different choices of the set of state variables.

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