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Today I was asked what does it mean for a physical property of a system to be intensive.

My first answer, loosely speaking, was:

"It is a property that is local."

I was specifically thinking about density and, by "local", I meant "that is unaffected by the dimension of the system". Ofcourse this is a very ambiguous answer, so after that I said (shifting to extensivity's definition):

"A property is extensive if it depends on the volume of the system observed."

To be honest, I said if it's proportional to the volume, but I'm not sure the this is correct. Now, that I'm still thinking about it, I've come to the conclusion that a good definition could be:

"A property is extensive if it depends on the quantity of matter of the system observed."

Looking on wikipedia I realized that this is exactly the definition given. But I'm somewhat still uncomfortable with that: if a gas is kept in a recipient of volume $V$ at a temperature $T$, his pressure is function of the number of moles of the gas:$$p=n(RT/V).$$ And, as we know, pressure is an intensive property. So (to me) it is not really clear what does "does not depend on the quantity of matter" mean.

I also thought that one could use an operational definition (if this is the good term) of extensivity/intensivity: one example might be:

"Suppose to measure a quantity $q(S)$ relative to a system $S$. Now reproduce a copy of $S$ and measure the same quantity for the system $S+S$ given by the two identicaly systems joined. If $q(S+S)=q(S)$, then $q$ is an intensive quantity."

This seems to give a more precise sense to the "does not depend on the quantity of matter" in the above definition, but there are gaps to fill. Maybe I will try to develop better this in a second time. Now, ofcourse, the question is: what is the definition of extensivity/intensivity in rigorous and unambiguous terms?

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Personally, your last example is exactly how I would define intensive quantities:

"Suppose to measure a quantity $q(S)$ relative to a system $S$. Now reproduce a copy of $S$ and measure the same quantity for the system $S+S$ composed of the two identical systems considered as a single system. If $q(S+S)=q(S)$, then $q$ is an intensive quantity."

I edited it only slightly, because it's important that the two identical copies of the system remain independent and non-interacting.

I would add to this that

If, for two different systems $S$ and $T$, $q(S+T)=q(S)+q(T)$, then $q$ is an extensive quantity.

Note that this does indeed mean that extensive quantities are proportional to the system's volume.

These two definitions leave room for quantities that are neither intensive nor extensive. That's OK - there are indeed many such possible quantities, although we don't use these terms to talk about them.

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    $\begingroup$ @Luke Somers (who suggested an edit): it would be great to have an example of a quantity that's not intensive or extensive according to this definition - but unfortunately resonant frequency doesn't work. The caveat that the systems have to be independent and non-interacting means that it would be intensive on this definition. $\endgroup$ – Nathaniel Sep 30 '13 at 15:08
  • $\begingroup$ How about the length of plane figures as an example of a quantity that is neither extensive nor intensive? The longest dimension of the unit square is $\sqrt 2$ and the longest dimension of a 2x1 rectangle is $\sqrt 5$. $\endgroup$ – David H Sep 30 '13 at 15:18
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    $\begingroup$ Variances add for independent random variables, so the standard deviation of any thermodynamic quantity grows as the square root of the system size - faster than an intensive quantity but slower than an extensive quantity. $\endgroup$ – tparker Jul 12 '17 at 15:16
  • $\begingroup$ Your definition is good, but I think your requirement that the subsystems do not interact is too restrictive, since in real life they usually do. I would tweak your definition to allow the subsystems to interact, but also specify that we only consider the asymptotic scaling behavior of large systems. $\endgroup$ – tparker Jul 12 '17 at 15:21
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    $\begingroup$ Some more complicated examples of natural physical quantities that grow subextensively are the entanglement entropy of the ground state of a quantum system, and arguably the entropy of a black hole. (The latter only grows with the surface area, not the volume, but that's a subtle case, because the volume inside of black hole, and therefore the "system size," can be tricky to define.) $\endgroup$ – tparker Jul 12 '17 at 17:31

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