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In Wikipedia, in the article about the statistical ensemble, it is said that in classical mechanics (thermodynamics and statistical mechanics) the ensemble is represented via the probability density in phase space. Is there an intuitive understanding as to how we make this assumption?

From a mathematical point of view, if you integrate the pdf of an arbitrary random variable to a region, you get the probability that the said variable, takes a value which is included in this region/interval (over which we are integrating). Adjusting that to our problem, the probability we get once we integrate the phase space pdf over a region, is the probability that our system occupies a microstate, that is within the region of integration. For simplicity I am considering the MCE as our main type of ensemble. I am not sure whether specifying that plays a huge role in what I am trying to understand, but just in case it does, I am specifying that.

My first question is:

How is the integral over a region translated for our statistical ensemble? The ensemble is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. I can't understand how our integration is translated in the abstract interpretation of the ensemble, like, what happens to the ensemble, when we integrate in phase space?

My 2nd question has to do with stationary statistical ensembles:

These are ensembles that do not evolve over time? What exactly is that one characteristics that doesn't change, or indicates that this ensemble is time independent? No change in the nr. of idealized virtual copies? No change in the number of the virtual copies being at a certain microstate? In other words, in the realm where the concept of the statistical ensemble exists (which I assume is not the realm of mathematics or that of the physical world), what it means for it to be time invariant. And most importantly, how this time invariance of an ensemble is reflected in phase space? Volume of the region over which we integrate and get probability 1, is constant but can change shape?

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A point in the phase space corresponds to a microstate of a system of interest.

The ensemble is the manifold of different possible microstates compatible with some external constraint.

To evaluate an average of some observable over a subset of microstates, we need a measure over the microstates, and the measure of the totality of microstates (the whole accessible phase space in that ensemble) must be finite. Therefore we can always get a probability measure. The celebrated Radon-Nykodym theorem ensures that all the different measures can be recast in the form of suitable probability densities.

Therefore, nothing happens to the ensemble when we integrate in the phase space. We just evaluate phase space averages that can be used in place of time averages in the case of ergodic systems.

An ensemble is characterized by its probability density. Generally, it is a function of the phase space point, the parameters required to represent a macrostate, and time. It turns out that time-independent densities imply time-independent averages. Therefore time independent densities are necessary to describe equilibrium macrostates.

However, this does not imply that there is no microscopic dynamics at equilibrium inside a single system. The individual microstates have their dynamics. But the resulting probability density does not change at equilibrium.

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  • $\begingroup$ By average of an observable you mean it's expectation value ? In classical mechanics we usually find that as $\int \hat X \rho(\vec q,\vec p,t)d^{3N}qd^{3N}p$ and in quantum mechanics $\langle X \rangle = Tr(\hat X \rho)$ right? Is this what you mean with we " a measure over the microstates" ? And what you mean with this: "and the measure of the totality of microstates (the whole accessible phase space in that ensemble) must be finite."? That the region of integration must be finite? $\endgroup$
    – imbAF
    Commented Feb 10, 2023 at 17:21
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    $\begingroup$ @imbAF average = expectation value. By measure over the microstate, I mean what in mathematics is defined as a measure: en.wikipedia.org/wiki/Measure_(mathematics) Finite measure means that $ \int \rho =$ a finite value. $\endgroup$ Commented Feb 10, 2023 at 17:44
  • $\begingroup$ Two more things: 1- So when we say we have an statistical ensemble in equilibrium, what does this implies for the ensemble itself, not the expectation values or something else. And 2nd, is it correct to say that the phase space density function= macrostate= ensemble ? $\endgroup$
    – imbAF
    Commented Feb 10, 2023 at 18:06
  • $\begingroup$ @imbAF 1- Equilibrium implies, as a direct consequence of Liouville's theorem, that the density probability must depend on generalized coordinates and momenta only through the Hamiltonian. 2nd: density function, macrostate, and ensemble, although not completely coinciding, are strongly coupled concepts. $\endgroup$ Commented Feb 10, 2023 at 21:51

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