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Perhaps one of the most important results of the whole of Classical Mechanics is that the volume occupied by an ensemble in the phase space remains constant in time. Another very interesting result is that it is also invariant under canonical transformations (as the Jacobian of the canonical transformation is generically unity).

Both these results compel me to think that there should be some extremely simple and deep physical meaning to the volume of the phase space. Intuitively, it seems to me that since the number of points occupied in the phase space is just the number of microstates of the ensemble, the volume of the phase space also denotes just the number of microstates of the ensemble up to some (not-so-clear-to-me) scaling factor. My question, thus, is that what is this scaling factor? Is it just arbitrary (i.e., to be decided by some choice of units etc.) or is there a definite scaling factor?

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    $\begingroup$ The scaling factor between phase space volume and microstates is $h$, which can be seen from the WKB approximation $\int p \, dx = n h$ or more generally by Weyl's law. I asked a somewhat related question here. $\endgroup$
    – knzhou
    Aug 28 '18 at 8:09
  • $\begingroup$ You get a lot of great stuff for free here. For example, the classical adiabatic invariant is the phase space volume, so this must correspond in QM to the number of microstates. But that determines the entropy, so a thermodynamic adiabatic process leaves the entropy invariant. $\endgroup$
    – knzhou
    Aug 28 '18 at 8:10
  • $\begingroup$ @knzhou Thanks for your comments your question is quite related and the discussion there seems helpful. And yes, I agree that this thing relates to a lot of results in statistical mechanics and quantum mechanics. $\endgroup$
    – Dvij D.C.
    Aug 28 '18 at 8:16
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The volume remains constant only for conservative systems. Once there is dissipation or loss, typical ensembles of initial conditions will tend to sets of zero volume, called attractors, which can be equilibria, periodic limit cycles, or quite often fractal sets associated with chaotic behavior.

what is this scaling factor? Is it just arbitrary

At least for quantum systems, this factor is not arbitrary, but usually taken to be $\sim\hbar^d$, where $d$ is the number of phase space dimensions. In this context, phase space volume may be linked to "conservation of information".

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  • $\begingroup$ But do these dissipative systems occur at fundamental scales? I think dissipations are always the appearance of our ignorance of a part of the actual whole system. If you take into account the system which receives the dissipated energy and you form an appropriate higher-dimensional phase-space then you can always convert the dissipative description into a description of a conservative system. $\endgroup$
    – Dvij D.C.
    Aug 25 '17 at 12:13
  • $\begingroup$ @Dvij, yes, locally at least (i.e., ignoring cosmology) energy is essentially conserved. My first paragraph mostly refers to the general classical mechanics case. $\endgroup$
    – stafusa
    Aug 25 '17 at 12:19

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