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Intuitively it's easy to accept that the usual variables like temperature, internal energy, etc. are 'macroscopic', but does there exist a formal definition of a macroscopic variable?

In other words, is there a clear way to separate the set of all observables (and functions of observables) on a system into ones we would describe as 'macroscopic' and ones we would not?

EDIT: Since apparently the answer is not completely straightforward, I'm interested in hearing any definitions which have appeared in literature, even if they are only conventions. I'm also interested in any necessary or sufficient conditions.

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Let me focus on the context of rigorous equilibrium statistical physics (see, e.g., Georgii's book "Gibbs Measures and Phase Transitions"). There, one works with probability measures on infinite systems, often on a lattice; let me assume it's $\mathbb{Z}^d$. In this context, a macroscopic observable is defined as a function $O:\Omega\to\mathbb{R}$ (where $\Omega$ is the set of all configurations $(\omega_i)_{i\in\mathbb{Z}^d}$) which does not depend on the values of any finite sets of spins $\omega_i$ (technically, one says that such an observable is measurable with respect to the tail $\sigma$-field).

Let me give you some examples of such observables, in the simple case of Ising-type systems, i.e., with $\Omega=\{-1,1\}^{\mathbb{Z}^d}$. Let $\sigma_i$ denote the spin at $i$, $\sigma_i(\omega)=\omega_i$.

$\circ$ Averages of local observables, e.g., $$ \lim_{\Lambda\uparrow\mathbb{Z}^d} \frac1{|\Lambda|}\sum_{i\in\Lambda} \sigma_i\;. $$ $\circ$ Events such as "There are no infinite connected components of $-1$-spins".

In both cases, changing a finite number of spins does not modify the value of the observable $O$.

One nice thing about this definition is that one can prove very generally that such observables take deterministic values (i.e., are almost surely constant) with respect to any pure phase (extremal Gibbs measure). In other words, they do not fluctuate (remember one deals here with infinite systems).

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"changing a finite number of spins does not modify the value of the observable" - consequently there is no such thing as a macroscopic variable for a physical system according to this definition, since all physical systems contain only a finite number of degrees of freedom. – Nathaniel Mar 4 '13 at 8:44
@Nathaniel: Well, that's a simplistic point of view ;) . You should take the idealization consisting in replacing a huge finite system by an infinite one to be of the same type as what is often done in probability theory: for example, the strong law of large numbers requires an infinite sample. This is mostly for convenience: considering infinite systems gives the possibility to have clear-cut definitions; defining, say, phase transitions in a finite system leads to ambiguities. Of course, one can then complement this analysis with that of finite-size effects. – Yvan Velenik Mar 4 '13 at 9:09
The thermodynamic limit is a useful approximation, but it is nevertheless an approximation. Macroscopic variables, on the other hand, are a concept that we regularly apply to real, physical systems and not infinite-limit idealisations. – Nathaniel Mar 4 '13 at 9:22
So this then seems to imply the problem of how to map observables on infinite systems to observables on finite systems, right? – Ben Aaronson Mar 4 '13 at 9:27
@Nathaniel: Sure. Nevertheless, this is the notion that is used in this context. Of course, the infinite-volume claims that you can extract for such observables do have finite-volume counterparts (that you can rigorously derive) for their finite-volume, physically more directly relevant, versions. But, again, you lose the possibility of having clear-cut definitions. Note also that if you're interested in deriving thermodynamical properties of the system, then these approximations are very natural: in thermodynamics, one usually assumes that the shape of the sample is irrelevant [to be cont.] – Yvan Velenik Mar 4 '13 at 9:30

It's hard to give a precise definition because to some extent these terms are a matter of convention. A good-but-not-quite-correct starting point is to think of a macroscopic variable as being such that, if you know its value (i.e. you've measured it), the entropy of the system can still be non-zero. For quantum systems this is the von Neumann entropy, whereas for classical systems it is the usual Gibbs-Shannon entropy. A microscopic variable is one for which this is not the case - knowing its value completely determines the system's state in the classical case, or puts the system into a pure state in the quantum case.

This seems like a good definition because statistical mechanics is all about the cases where the microscopic details are unknown and have to be represented with a probability distribution or density matrix, and this definition seems to capture exactly those cases.

Note that this definition of "macroscopic" includes the expectation values of microscopic variables, since just knowing the expectation over an ensemble still leaves some uncertainty about the microscopic variable's precise value. The "extensive" macroscopic variables in thermodynamics ($U$, $V$ etc.) are of this kind.

However, the reason this isn't quite right is that in the case where there is some degeneracy in the energy levels, knowing the precise value of the energy can still leave a non-zero von Neumann entropy. But energy is generally considered a microscopic variable even in this case. An improvement to this definition might involve saying that a spanning set (my term) of microscopic variables is one such (a) they can be measured simultaneously (i.e. they're measured by orthogonal projection operators), and (b) knowing the values of all of them leaves zero von Neumann entropy. So, for example, knowing both the energy and the angular momentum of an orbiting electron might completely determine the wavefunction, and thus they would be a spanning set, even though each alone might not be. The question then would not be whether a single variable is a microscopic or macroscopic one, but whether a given set of variables is a spanning set or not. This seems like the right approach to me, but I don't know whether anyone has formally described it. (I would be interested to know if anyone has.)

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This is an interesting idea, but on its own it seems like a relatively weak condition. For example for a 1D chain of particles which may be oriented either 'up' or 'down' (let's just say classical for simplicity), you could know the exact orientation of every particle but the first and have nonzero entropy. Perhaps a more complete extension would be that it leaves the entropy of any arbitrary subsystem nonzero, but I'm still not sure how good a condition that is. – Ben Aaronson Mar 4 '13 at 3:35
@Stereotomy I guess I'm ok with it being a weak condition. I'd be ok with calling all-spins-but-one a macroscopic variable, because even in that simple case you have to use stat mech to work out its expected behaviour. (That's assuming the spins interact deterministically. If there's a heat bath then you'd have to do that anyway, so I'd even call all-the-spins-but-not-the-thermal-degrees-of-freedom macroscopic). Of course that isn't the sort of thing people usually mean to include when they say "macroscopic", but I think that's really just a matter of convention. – Nathaniel Mar 4 '13 at 4:39
To give a more nuanced definition, you could say that if you can factor the system's phase space into subsystems then it's OK to talk about the microscopic variables of a specific subsystem. So all-spins-but-one would be a (spanning set of) microscopic variable(s) of the subsystem composed of all particles but one, but it's not a microscopic variable of the system composed of all the particles. Does that make sense? – Nathaniel Mar 4 '13 at 4:44
Sure, that makes sense, and I understand that it's a matter of convention. I've edited the original question to broaden it somewhat. I'm still interested to know if anybody has attempted to answer this in literature, even if only to precisely define a convention. – Ben Aaronson Mar 4 '13 at 4:51
State dependence is another issue here. For example if you start with a 0 entropy state, then all variables become macroscopic. To me it feels like it's not the zero-ness of entropy so much as the exponential drop in entropy that intuitively feels important. I have no idea how one would make that rigorous though. – Ben Aaronson Mar 4 '13 at 9:22

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