Fundamental relationship between thermodynamics and stat. mechanics

[Daaavde]

From the Greiner (Thermodynamics and statistical mechanics) on the relationship between the number of microstates of two systems and the total entropy:



...for two statistically independent systems the total number of compatible microstates $\Omega_{tot}$ is obviously the product of the numbers for the individual systems, namely $\Omega_{tot} = \Omega_1 \Omega_2$. We have seen the entropy is an extensive quantity which is simply added for both partial systems: $S_{tot} = S_1 + S_2$.
If we now assume that there is a one-to-one correspondence between entropy and $\Omega$, for instance $S = f(\Omega)$, there is only one mathematical function which simultaneously fulfills $S_{tot} = S_1 + S_2$ and $\Omega_{tot} = \Omega_1 \Omega_2$: the logarithm. Therefore it must hold that $S \propto ln \Omega$...



Now, in this case, the one-to-one correspondence between $S$ and $\Omega$ is not obvious to me at all. I was searching for a demonstration, or at least some more convincing justification, but i found nothing.

Can anyone help me with this?

 

[Andrew Mason]

From the Greiner (Thermodynamics and statistical mechanics) on the relationship between the number of microstates of two systems and the total entropy:



...for two statistically independent systems the total number of compatible microstates $\Omega_{tot}$ is obviously the product of the numbers for the individual systems, namely $\Omega_{tot} = \Omega_1 \Omega_2$. We have seen the entropy is an extensive quantity which is simply added for both partial systems: $S_{tot} = S_1 + S_2$.
If we now assume that there is a one-to-one correspondence between entropy and $\Omega$, for instance $S = f(\Omega)$, there is only one mathematical function which simultaneously fulfills $S_{tot} = S_1 + S_2$ and $\Omega_{tot} = \Omega_1 \Omega_2$: the logarithm. Therefore it must hold that $S \propto ln \Omega$...



Now, in this case, the one-to-one correspondence between $S$ and $\Omega$ is not obvious to me at all. I was searching for a demonstration, or at least some more convincing justification, but i found nothing.

Can anyone help me with this?

The one-to-one correspondence is just an assumption. It is not obvious so they just ask you to assume it. It took Boltzmann years to develop the relationship between entropy and Ω, so don't feel bad if you are having difficulty seeing the connection.

AM

 

[Daaavde]

I would be ok with the ansatz (it's not the first time I've to deal with it of course) but my doubt is:

The assumption is just confirmed by experimental result (hence we're like "yeah, data are fitting so Boltzmann is right, let's leave it like this") or there is a formal demonstration about the bijective correspondence S-Ω (i hope so) which maybe is too complicated (or too abstract) for the Greiner to put in the book.

I don't necessarily need the demonstration (though I would really like to give it a look), I would just like to know if the assumption is somewhere formally justified or not.

 

[Jano L.]

One way to show it is to derive that the quantity defined by

$$
S_{stat} = k_B \ln \Omega
$$

behaves in the same way as the thermodynamic entropy $S$. Some subtle points are encountered, like how to count number of states properly but the simple version is given in many textbooks on statistical physics.