Undergrad Extensive properties as measures

[burakumin]

It has always struck me that extensive quantities (kinetic energy, volume, momentum, angular momentum, mass, entropy, ...) could be defined as measures (https://en.wikipedia.org/wiki/Measure_(mathematics)) whereas intensive quantities are fields. Are there known ressources that put emphasis on this aspect? In particular I'm curious about how forces and potential energy have be handled.

 

[andrewkirk]

It would not work for quantities such as momentum and angular momentum because they have vector values whereas a measure has a real scalar value. One can replace a vector momentum by the scalar component of the momentum in a given direction, but that will then fail the non-negativity criterion required of a measure. It would be reasonable to say that positive-scalar-valued extensive quantities - which I think includes all the thermodynamic ones - are measures.

It seems reasonable to think of an intensive quantity as a scalar field up to a point, since in most cases the quantity is of a ratio of some quantity to as volume. But the analogy breaks down when the volume gets very small. For instance temperature is related to the average KE per molecule, so once we have a volume smaller than a molecule it no longer makes sense. Fields need to be defined at every point in space whereas intensive quantities are defined for positive volumes, which can be very small, but cannot be points.

 

[burakumin]

It would not work for quantities such as momentum and angular momentum because they have vector values whereas a measure has a real scalar value. One can replace a vector momentum by the scalar component of the momentum in a given direction, but that will then fail the non-negativity criterion required of a measure. It would be reasonable to say that positive-scalar-valued extensive quantities - which I think includes all the thermodynamic ones - are measures.

You can define signed measures, vector measures, projection-valued measures, ... The concept is not limited to positive scalar valued things. The only advantage of standard measures is that they can naturally deal with infinity whereas others cannot in a simple way (but this is hardly a crucial concern in physics). But for example I've never found any good argument of why it would not be a good idea to define momentum as a vector measure.

But the analogy breaks down when the volume gets very small.

Sure but this not really my point here. Continuity of matter is pervasive in many domains of physics even if it is only an approximation.

My question was more about quantities that are neither extensive nor intensive.