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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.

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    $\begingroup$ IMO your question is totally unclear. $\endgroup$ Jul 4, 2022 at 15:37
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    $\begingroup$ What is a closed form of thermodynamic variables/ equation of state? $\endgroup$ Jul 4, 2022 at 16:08
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    $\begingroup$ What does it mean guarantee closed form of thermodynamic variables (equation of state) for an arbitrary system? We usually can express pressure and the functions of state in terms of parameters of the system (as long as we have its Hamiltonian)... but not necessarily in a simple or even mathematically tractable equation. $\endgroup$
    – Roger V.
    Jul 4, 2022 at 16:13
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    $\begingroup$ It is about how many parameters you need to describe a distribution (see Sufficient statistics). The trick in stat mech is that we treat positions and momenta as random variables, distribution of which is parametrized by very few parameters, because the particles are identical. We need parameters that determine the configuration of the system (pressure, magnetic field, etc.) + temperature. Perhaps, you could try to reformulate your question more clearly... but now, that I understand you, I think it is an interesting one. +1 $\endgroup$
    – Roger V.
    Jul 4, 2022 at 16:23
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    $\begingroup$ I think your question boils down to, "when can I write the equation of state for a system in closed form?" To which I suspect the correct answer is "only in very special ideal systems." In practice, I think what is done is to make more or less educated guesses about "reasonable" equations of state and fit them to empirical data. There's not a unique way to do this, and different methods/guesses/approximations will have different advantages and disadvantages, so that's why you can find many equations of state in the literature for nuclear matter, for example. $\endgroup$
    – Andrew
    Jul 4, 2022 at 20:48

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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.

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.

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