# Is there a formal definition of a macroscopic variable in statistical mechanics?

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 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 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 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? – Stereotomy Mar 4 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 at 9:30
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.