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Imagine a small amount of gas in a container. Would it theoretically be possible to measure the positions and momenta from all the gas particles? Would there be a hard limit? What would the barriers be?

Would there also be a difference in the case the particles would act classically? (because the hard limit would be obviously heisenbergs indeterminacy relation)

Basically I am trying to understand how small you can theoretically make the phase space volume in $\Gamma-$space.

I am also wondering if the problem related to the Gibbs paradox prevents from knowing all the positions and momenta of the gas particles.

Since the particles are indistinguishable, it seems impossible to make a list like this:

  1. particle $1$ has $q_1$ and $p_1$
  2. particle $2$ has $q_2$ and $p_2$ ...

Since we cannot label the particles as particle $1$ and particle $2$..

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  • $\begingroup$ How do you suppose to measure even the state of 3 particles in your volume? $\endgroup$
    – trula
    Commented Sep 19, 2023 at 9:51
  • $\begingroup$ I don't see why indistinguishability would be fundamental obstacle. You could either pick an arbitrary labeling or store the states in an unordered data structure. $\endgroup$
    – leapsheep
    Commented Sep 22, 2023 at 10:40
  • $\begingroup$ Well imagine measuring the properties of a single atom in ideal gas. Then use that to predict classically where it would be at time $t$. At that moment measure that particle again. If you would find the same particle, it wouldn't be indistinguishable in the first place $\endgroup$
    – bananenheld
    Commented Sep 22, 2023 at 11:13

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Not sure where you are going with this, but however you cut it, the phase space of the gas is going to mind-bogglingly huge. Even if you treat molecules as point particles, each particle needs $6$ dimensions in phase space - three for its position and three for its momentum. If you take into account the rotation of each molecule and its internal vibrations then it needs even more dimensions. And you have around $10^{22}$ molecules in even a small amount of gas - say one litre. So that's a phase space with around $10^{23}$ dimensions. And that's before you even start thinking about the precision of the measurements.

Even if you only measure each of the $10^{23}$ parameters of the gas with a precision of $10$ bits (about one part in a thousand), you now have $10^{24}$ bits of data. That is a yottabit of data, which is rather more than all of the digital data in all formats in the whole world.

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  • $\begingroup$ Thank you for taking the time to answer. I see, it would take an enormous amount of data to register the measurements. But I am guessing that it would even be theoretically impossible, because any measurement would basically make the information you already gathered useless right? Even if you would have a huge memory disk $\endgroup$
    – bananenheld
    Commented Sep 19, 2023 at 12:21
  • $\begingroup$ @bananenheld If you ignore all practical issues involved in making the measurements, storing the data etc. then the only theoretical constraint is Heisenberg's Uncertainty Principle. $\endgroup$
    – gandalf61
    Commented Sep 19, 2023 at 17:30
  • $\begingroup$ How are you sure that that is the only theoretical constraint? It seems to me that it is impossible to gather all the information to the limit given by heisenberg's uncertainty principle, but I cannot articulate why. This is my central question $\endgroup$
    – bananenheld
    Commented Sep 20, 2023 at 9:47

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