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I've been reading Landau and Lifshitz book on Statistical Mechanicals and some aspects about how they lay out the principles has me a little confused.

Let us now consider a macroscopic body or system of bodies, and assume that the system is closed, i.e. does not interact with any other bodies. A part of the system, which is very small compared with the whole system but still macroscopic, may be imagined to be separated from the rest; clearly, when the number of particles in the whole system is sufficiently large, the number in a small part of it may still be very large. Such relatively small but still macroscopic parts will be called subsystems. A subsystem is again a mechanical system, but not a closed one; on the contrary, it interacts in various ways with the other parts of the system.

So firstly I believe that “closed” in the first sentence means isolated (from “does not interact with other bodies”). They go on to say that we will be analyzing subsystems of a given system. Why cant we just cut out the middleman and look at the system as a whole? It seems that everything they lay out should apply if we were to study the system as a whole. It seems like eventually we essentially do this (when we sum over the subsystems of the system) but for some reason we initially narrow our focus to a specific subsystem.

Also, he goes on to say that the system will occupy all of phase space at some point. Any realistic system will be bound (usually in space by walls), making this impossible. Does this have any ramifications on what we can say about the system?

I have a lot of questions and some ideas but the first question is definitely the most fundamental and I dont want to just post a list of questions. Any help appreciated.

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  • $\begingroup$ Dividing the system up into sub-systems is used as part of a mechanism of finding the system states with maximum entropy when at equilibrium. $\endgroup$
    – Farcher
    Commented Jun 10 at 8:32
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    $\begingroup$ While I have used and loved multiple L&L books, expect them to be concise rather than clear. Landau seemed to assume that you knew as much as he did, which certainly wasn't the case for me. $\endgroup$
    – Jon Custer
    Commented Jun 10 at 14:56

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Let's go through your questions in sequence:

So firstly I believe that “closed” in the first sentence means isolated

This is correct.

Why cant we just cut out the middleman and look at the system as a whole?

This gives a clear way of connecting the microcanonical ensemble (here, the system) to the [grand] canonical ensemble (here, the subsystem), in precisely the stated limit of the system being much larger than the subsystem.

In this way, we can apply statistical results we know from the microcanonical ensemble to figure out properties of the subsystem. In applications, we often want to consider some (sub)system in equilibrium with a much larger bath (e.g. the environment), so it is useful to study properties of such a subsystem.

Also, he goes on to say that the system will occupy all of phase space at some point. Any realistic system will be bound (usually in space by walls), making this impossible. Does this have any ramifications on what we can say about the system?

This is essentially the statement of the ergodic hypothesis. Of course, the system may explore all of the available phase space after sufficient time, which excludes the inaccessible regions of phase space.

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  • $\begingroup$ I looked at these slides to help understand some of the points you made. When I think of a subsystem I think of drawing a box in our system (usually a gas in a box) and analyzing the particles in the box at any instant. Your comment and the slides seem to imply that we can have systems in equilibrium with statistical properties that vary among subsystems. I guess I am confused as to how this can actually be achieved (1) $\endgroup$
    – user62783
    Commented Jun 10 at 16:43
  • $\begingroup$ Because it seems to me that the only way to achieve a non-homogenous system is to “seperate” the subsystem is to introduce walls/a container inside the system which seems like it would prevent the ergodic hypothesis from being true (the particles constituting the walls won’t be able to obtain significant momenta—im not sure if this ties in to what you clarified about only the AVAILIBLE phase space being relevant). Maybe my understanding of a “system” is flawed—is it okay if we have matter of various states (solid,liquid,gas), that interact with each other (I.e. bonds in a solid) $\endgroup$
    – user62783
    Commented Jun 10 at 17:05

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