statistical mechanics - Limitations of the ideal gas law: When does it fail?

I ask this question precisely because I am looking for a fundamental, quantitative explanation of the limitations of the ideal gas law and when it should not be used.

Note, I have found the following answers https://physics.stackexchange.com/a/91727/59023, https://physics.stackexchange.com/a/43701/59023, and https://physics.stackexchange.com/a/17364/59023 but in each case there was not a definitive or quantitative limitation prescribed. The answers mention the limitations but it is not clear whether those limitations invalidate the use of $P = n \ k_{B} \ T$ [i.e., $n$ is the number density, $k_{B}$ the Boltzmann constant, and $T$ a scalar temperature (well, one could make it a pseudotensor if they prefer)].

My primary motivation in asking this question derives from a conversation I had with a colleague about the differences between equilibrium and nonequilibrium gases.

For a specific example, consider the nearly collisionless (well, weakly collisional at best) plasma in the solar wind. Here all the interactions are due to long-range forces (i.e., Coulomb collisions and/or interactions with fluctuating electromagnetic fields), which as several of the answers above eluded, is a condition that the ideal gas law assumes to be negligible. Further, the nearly collisionless, nonequilibrium state of this ionized kinetic gas coupled with the ubiquitous observations of non-Maxwellian, anisotropic velocity distributions (e.g., see What is the correct relativistic distribution function? and references therein) causes me to doubt the applicability of ideal gas to these plasmas.

However, I know of several examples in physics where the assumptions used to derive relationships/approximations like the ideal gas law are not rendered invalid simply because one or more of the assumptions do not hold.

To be clear, I am not concerned with:

• the kinetic definition of pressure; or


• the kinetic definition of number density; or


• about how one defines a kinetic temperature since we can define what is meant by $T$ as @AccidentalFourierTransform eludes to in comments to the question at Derivation of ideal gas law.

• I am curious how poor is the approximation $P = n \ k_{B} \ T$ for the solar wind plasma?
  
  
  
  • Note: I am not concerned about the multiple species or components so much as the underlying reasons justifying when we can assume $P = n \ k_{B} \ T$.
  
  
  
  


• Should we consider a different equation of state for the system to derive a new/different relationship between pressure, density, and temperature?


• Which is more critical to the relevance of the validity of $P = n \ k_{B} \ T$: the nonequilibrium or nearly collisionless state of the plasma?