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In classical statistical mechanics we have to divide the partition function by a factor of $1/h^n$. In almost every calculation of a real quantity this cancels out and is thought to be a remnant of quantum mechanics.

However in the grand canonical ensemble we find in some rare equations the persistence of this term in the final result. For instance the Saha equation as derived from classical statistical mechanics has this factor remaining.

Perhaps related to this is the way we define a probability distribution on the phase space. How do we define (in terms of symplectic manifold theory) a phase space with variable number of particles (and hence dimensions)?

Does this lead to the weirdness that we are observing in some results of classical statistical mechanics retaining the $h$ factor? In essence does particle creation and disappearance violate formulation of mechanics on a manifold.

AngusTheMan
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    The number of available physical states per unit volume of phase space per particle factors out if the total number of particles is conserved. If not then the result will depend on this number. But note that the very reason why you end up with a changing number of particles is an artifact of invoking quantum mechanics, e.g. neutral hydrogen consists of a proton and an electron in a bound state but you are not treating this classically where you would have a continuum of states, rather you take into account that the electron is frozen in the ground state. – Count Iblis Aug 23 '15 at 21:02
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    There is no problem in defining infinite dimensional manifolds with symplectic structures, and hence configuration and phase space. Also probability measures (distributions) are allowed in infinite dimensional space (provided they have locally convex topology). – yuggib Aug 23 '15 at 21:32
  • Thank you both for your comments. In this case, can we view classical statistical mechanics as a Hamiltonian field theory? – AngusTheMan Aug 24 '15 at 10:19

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