A fundamental relation contains all the information about a system--all the thermodynamic information about a thermodynamic system--and involves independent extensive variables. For instance, entropy S expressed as a function of internal energy U, system volume V, and conserved particle number N (same type of particles) is a fundamental relation, expressed symbolically as S(U,V,N). Or, equivalently, the internal energy U can be expressed as a function of entropy, volume, and particle number, U(S,V,N), and this is also a fundamental relation.
In contrast, the more commonly used equations of state are not fundamental relations, and involve functions of state that are not independent of each other, such as temperature T, pressure P, and chemical potential μ, that are partial derivatives of fundamental relations. A complete set of equations of state, however, is informationally equivalent to a fundamental relation. The three equations of state
T(S,V,N) = ∂U/∂S,
-P(S,V,N) = ∂U/∂V,
and μ(S,V,N) = ∂U/∂N,
for example, contain all the information of the energetic fundamental relation, U(S,V,N). For an interesting case where N isn't involved (because it's not conserved) see my July 22 post on the Stefan-Boltzmann law as one of the equations of state for black-body radiation.
(This discussion should make you think of similar descriptions used in quantum theory, such as "the wave function contains all the information about the system," the need for "a complete set of commuting observables," and the choice of energy or entropy fundamental relations being somewhat like the choice of Schrödinger or Heisenberg 'pictures'.)
The connections between fundamental relations and equations of state are discussed in Callen's book in the first three sections of Chapter 3, "Some Formal Relationships and Sample Systems." In case you don't have access to that, I'm posting (below) some meticulous notes on this subject put together for a Statistical Physics class by Professor Mark Loewe in the mid-1990s, when he was teaching at Southwest Texas State University in San Marcos (now Texas State University-San Marcos). He used Callen's book, but he also handed out notes such as these that he wrote himself. He is concise and thorough, as you can see. (Well, yeh, sorry the notes are hard to see. Slide 'em over to your desktop, maybe.) There's more info in the notes than you would likely ever want or need, but the accompanying descriptions are worth reading. One little thing Mark wasn't thorough about: in the corrections and clarifications, he doesn't say which problem Callen gives an incorrect answer to. Maybe I have it in my class notes, and will post it if I find it.
Just how fundamental relations and equations of state are related to the first law of thermodynamics in its most basic form,
,
,is something worth thinking about and looking up. Does the first law contain all the thermodynamic (macroscopic) information about a system? Oh yeah, and we have to talk about isolated systems versus systems attached to a heat reservoir. The 1st law equation above is for the change in internal energy for Q (energy as heat) supplied to the system, and W the work done by the system on its surroundings. It's a good starting point for imagining either an isolated system that has energy added to it (temporarily connected to, then disconnected from, some kind of reservoir) or a system connected to a heat reservoir and not yet in thermal equilibrium with the reservoir.
Just to keep things in perspective before going on to specific examples in later posts, I'll quote Callen from the top of page 26, "The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system."
Mark Loewe's notes:
