Using Homogenuous Functions to Understand Thermodynamics

[Petar Mali]

If I have homogenuous function $$f(x,y,z,...)$$ of degree $$r$$ than:
$$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+...=rf$$

In thermodynamics:
$$dU=TdS-pdV+\mu dN$$

If I said U is homogenuous function of degree 1 I will get

$$U=TS-pV+\mu N$$

When can I use this assumption?

 

[LeonhardEuler]

You can almost always use this assumption. It amounts to the assumption that when you have some substance, and you double the amount of it, and so its volume and entropy, you also double the energy. This will be true for macroscopic amounts of a substance.

The only exceptions occur for very small amounts of a substance. If you have one molecule, and then you add one more, the energy is not just doubled. Same goes for going from two molecules to four. It will start working when there is enough of the substance that the effects of the surface are negligible and you basically have a totally homogeneous material.

 

[Count Iblis]

You also have to assume that there are no long range interactions between the molecules. So e.g., in astrophysical problems where gravity is important you cannot make this assumption.

 

[Petar Mali]

Thanks!

 

[Tobiasz]

Hi!
You know..? This is exacly what I need for my thesis.
In which mathematic books can I find that theorem?