# 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.

• IMO your question is totally unclear. Jul 4, 2022 at 15:37
• What is a closed form of thermodynamic variables/ equation of state? Jul 4, 2022 at 16:08
• 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. Jul 4, 2022 at 16:13
• 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 Jul 4, 2022 at 16:23
• 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. Jul 4, 2022 at 20:48