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The microcanonical ensemble is usual motivated solely by the principle of indifference. Textbooks usually say something along the lines of "If the only thing we know about a system is its total energy, then all microstates with that energy should be considered equally likely. Therefore, for a system with N generalized coordinates $q_i$ and conjugate momenta $p_i$ with Hamiltonian $H(\{q_i, p_i\})$ and total energy $E$, the expectation value of a quantity $X(\{q_i, p_i\})$ is

$\langle X \rangle \propto \prod_i \int dq_i dp_i\, X(\{q_i, p_i\})\ f( H(\{q_i, p_i\}) - E)$

where $f$ is some function narrowly peaked about $0$, e.g. a delta function." (E.g. https://en.wikipedia.org/wiki/Microcanonical_ensemble#Classical_mechanical and https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)#Classical_mechanical)

But it seems to me that this equation makes a much stronger claim than can be justified solely the principle of indifference, because it's assuming that appropriate measure on phase space is the symplectic volume form. This is certainly a nice measure (e.g. it's invariant under canonical transformations), but one could easily imagine other measures. Is there a way to derive (or at least motivate) that the symplectic volume form is the correct measure on phase space in the context of statistical mechanics, or is this an additional nontrivial postulate contained within the fundamental postulate of classical statistical mechanics?

Does the answer hinge on Liouville's theorem? The theorem seems to provide a strong case for the symplectic volume form measure (by guaranteeing that it's conserved under time evolution). But I've never heard a professor invoke Liouville's theorem to justify the canonical ensemble probability density function, so if it is necessary, then this fact definitely gets glossed over in most elementary treatments.

(I know that in quantum mechanics, it's much easier to argue that the natural unit of phase space is $dq_i dp_i/h$, but I'm curious if you can justify this measure solely in the classical context.)

Edit: I suppose that if you parameterize your system using the phase-space coordinates $\{ q_i, p_i \}$ of Hamiltonian mechanics, then the symplectic volume form is the natural one to use. But how do we know that this is the appropriate parameterizion of the state space of our system for doing statistical mechanics? For example, in Lagrangian mechanics the fundamental degrees of freedom are instead $\{ q_i, \dot{q}_i \}$.

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  • $\begingroup$ related: (51927) $\endgroup$ – Abhinav Apr 23 '16 at 2:56
  • $\begingroup$ The Wikipedia article on "Principle of indifference" says ... $\endgroup$ – tparker Aug 14 '16 at 17:22
  • $\begingroup$ "The fundamental hypothesis of statistical physics, that any two microstates of a system with the same total energy are equally probable at equilibrium, is in a sense an example of the principle of indifference. However, when the microstates are described by continuous variables (such as positions and momenta), an additional physical basis is needed in order to explain under which parameterization the probability density will be uniform. Liouville's theorem justifies the use of canonically conjugate variables, such as positions and their conjugate momenta." $\endgroup$ – tparker Aug 14 '16 at 17:22
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Great question. I believe that yes, Liouville's theorem is the key part of the justification for this in classical stat mech.

The reason for this is that it leads to a time-invariant equilibrium measure. If you used a volume measure that wasn't time-invariant in this way then it would be very strange, because on the one hand you would say that you had no knowledge about the state of the system (besides its energy) at time $t_0$, but on the other hand your knowledge about the system at time $t_1>t_0$ would be different even without measuring it. That is, integrating forward in time would give a different probability density function, which in general would have a different entropy, either higher or lower.

I suppose that for many systems there are probably multiple other possible volume measures that are also time-invariant. However, the simplectic form is unique in being time-invariant for all closed systems, and if your system might interact with another system in the future then it is necessary to use that one for consistency.

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