# Extensive properties as measures

• I
• burakumin
In summary, the conversation discusses the concept of extensive and intensive quantities and whether they can be defined as measures or fields. It is argued that while positive-scalar-valued extensive quantities can be considered measures, quantities like momentum and angular momentum, which have vector values, cannot. The possibility of defining momentum as a vector measure is also mentioned. The conversation also touches on the limitations of using fields to describe intensive quantities and the issue of continuity in physics. The question is raised about how to define quantities that are neither extensive nor intensive.
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.

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.

andrewkirk said:
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.

andrewkirk said:
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.

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## 1. What are extensive properties?

Extensive properties are physical properties of a substance that depend on the amount or size of the substance. They are directly proportional to the size or amount of the substance and change when the size or amount changes.

## 2. How are extensive properties measured?

Extensive properties are measured using various units of measurement such as mass, volume, and energy. The unit of measurement used depends on the specific extensive property being measured.

## 3. What is the difference between extensive and intensive properties?

The main difference between extensive and intensive properties is that intensive properties do not depend on the amount or size of the substance, whereas extensive properties do. Intensive properties are constant regardless of the amount or size of the substance.

## 4. Can extensive properties be additive?

Yes, extensive properties can be additive. This means that the total value of an extensive property for a mixture of substances is equal to the sum of the individual values for each substance in the mixture. For example, the total mass of a mixture is equal to the sum of the masses of its individual components.

## 5. How do extensive properties relate to the concept of matter?

Extensive properties are an integral part of the definition of matter. Matter is anything that has mass and takes up space, and extensive properties are physical properties that allow us to measure and quantify these characteristics of matter.

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