# Can you Have a Macrostate of Position?

In Statistical Mechanics it seems a Macrostate is normally tied to some measurable quantity like Energy, or Temperature.

But is it possible to have a Macrostate of Position? For example, if I have a single unitcell of some crystal structure and within that unitcell there are 10 dopant positions, would it be reasonable to suggest that adding a single dopant, and it being in any one of the possible positions is actually a Macrostate of Position associated with some large number of possible microstates?

More generally, could a crystal structure be thought of as a Macrostate of Position?

Normally in statistical mechanics and thermodynamics we assume the center of mass position of the system is at rest, since an overall average motion of the entire system is usually a distraction from studying the internal properties of the system. So this kind of position is not really part of the macrostate since it doesn't change.

In the context of Lagrangian fluid mechanics, you can consider a fluid element which is "small" compared to the entire volume of fluid but "large" in the sense that it is made of many particles, and ask how the position of this fluid element evolves with time. Since the position of the fluid element is an average of many particles, and therefore subject to statistical fluctuations, it is a macroscopic variable. I'm not sure, however, if it's useful to think of the position of a fluid element as part of the "macrostate" of a thermodynamic ensemble; my guess would be no... since you rarely are directly measuring the positions of individual fluid elements, and since in an incompressible flow there is no heat transfer at all. (However this isn't my area of expertise).

In the example given in your question, you are talking about the position of one microscopic doping site in a single microscopic unit cell of a crystal. I would consider the position of the dopant to be part of the microstate, and typically not something that is macroscopically observable. A macroscopic variable might be more like the fraction of all doping sites in the crystal that contain a dopant.

• I suppose the question is, when are there enough microstates associated with some property for it to be considered a macrostate? Or, if the state is hard to measure can it ever be considered a macrostate? Would it then be something like a quasi-macrostate? Oct 23, 2021 at 20:54
• @Connor I tend to think of the microstates as being the truly fundamental quantity, and the macrostates as something we come up with because they are useful for us. We can measure the pressure and temperature, so it's useful to use these to label macrostates. At a deeper level, some properties of a system are transient or change on a rapid time scale, and others (which tend to be associated with conservation laws) are more stable; the macrostates tend to correspond to fixed values of these slowly varying quantities. But, that's because these are the quantities stable enough for us to measure Oct 23, 2021 at 20:57
• So there's not a hard boundary for whether something "counts" as a macrostate. On the other hand, if you choose macrostates that are too fine-grained, you may have to work a lot harder than you need to. Oct 23, 2021 at 20:58
• I tend to think of them in the same way you do, i.e. a macrostate is some collection of microstates with a defined property in common. I suppose here we are conserving structure, would that be topological conservation? Either way, to avoid confusion I may well just use quasi-macrostate. Oct 23, 2021 at 21:04
• @Connor Depending on your level, you might be interested in Section 2.4 of David Tong's notes on kinetic theory: damtp.cam.ac.uk/user/tong/kinetic.html. He shows that typical macroscopic dynamic variables like density, temperature, and velocity can be understood as quantities that vary on slow time scales (and so are stable enough to be worth discussing). Oct 23, 2021 at 21:07