When can we guarantee a closed form of statistical mechanical equations? Consider the equation of state:
$$P = \rho k_BT$$
where $P$ is pressure, $\rho$ is number density, $k_B$ is Boltzmann constant and $T$ is temperature. All variables are intensive thermodynamic variables.
I don't use $2$ moles of variables to describe the (position and momentum) "state" of $1$ mole of gas. I use 3 (in the ideal gas law), which I find remarkable and begs the question, when is this possible?
When can we guarantee a closed form of thermodynamic variables (equation of state), for an arbitrary system like gas, plasma, solid, etc.?
I'm not sure when this can or cannot be done for a system.
 A: First, note that there is a fundamental difference between knowing the 2 moles of variables describing a one-dimensional classical system of 1 mole of particles and the three variables appearing in your equation of state. Whereas the microscopic description fully characterizes the system in all its complexity, the equation of state is related to a statistical description of the system. This is, you could "read" the macroscopic information from the 2 moles microscopic variables, whereas the inverse is not true.
The statistical description of a system with many components does not rely (as far as I know) on any condition. However, the use of the theory of equilibrium statistical mechanics (Maxwell-Boltzmann-Gibbs) to make such a statistical description of equilibrium steady states does rely on some requirements.

*

*The thermodynamic limit ($N\rightarrow\infty$) is needed in order to ensure the equivalence between ensembles and the irrelevance of fluctuations.

*The stability of the potential (potential energy lower bound) is needed in order to avoid "catastrophic configurations with infinite particles located in a finite space."

*Short range interactions (interparticle potentials decay with exponents greater than the dimensionality
of the embedding space) in order to ensure extensiveness of the thermodynamic potentials.

Apart from these general requirements, there is a non-trivial equation relating the pressure with the rest of the thermodynamic variables only when the partition function (supposing we are working in the canonical framework) depends explicitly on the volume.
