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?