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*...for two statistically independent systems the total number of compatible microstates [itex]\Omega_{tot}[/itex] is obviously the product of the numbers for the individual systems, namely [itex]\Omega_{tot} = \Omega_1 \Omega_2[/itex]. We have seen the entropy is an extensive quantity which is simply added for both partial systems: [itex]S_{tot} = S_1 + S_2[/itex].*

**, for instance [itex]S = f(\Omega)[/itex], there is only one mathematical function which simultaneously fulfills [itex]S_{tot} = S_1 + S_2[/itex] and [itex]\Omega_{tot} = \Omega_1 \Omega_2[/itex]: the logarithm. Therefore it must hold that [itex]S \propto ln \Omega[/itex]...**__If we now assume that there is a one-to-one correspondence between entropy and [itex]\Omega[/itex]__Now, in this case, the

**one-to-one correspondence**between [itex]S[/itex] and [itex]\Omega[/itex] 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?