Why does entropy obey the superposition principle?

Question

I was deriving the Boltzmann's entropy formula

$${\displaystyle S=k_{\mathrm {B} }\ln \Omega}$$

We start with two prepositions:

1. Let's consider two systems and we know the entropy of the first is $S_1$ and of the second $S_2$. If we want to know the entropy of both systems, it should be

$$S_{12}=S_1+S_2\tag1$$

2. The probability of a state of the first system is $p_1$ and of the second $p_2$. So the probability of the global state of those two systems is:

$$p_{12}=p_1\cdot p_2$$

If $S=f(p)$ than we should have a function:

$$f(p_1)+f(p_2) = f(p_1\cdot p_2)$$

from which by differentiating and integrating we obtain the Boltzmann's entropy formula.

How do we know that equation $(1)$ is true? Why does entropy obey the superposition principle? Is that something from classical thermodynamics or can it be shown only outside of it?

Answer 1

I think you are confusing the causes and consequences here. First of all, Boltzmann did not introduce the concept of Entropy $S$, Clausius did and it was an extensive additive property by definition (because it was defined through heat, itself an additive quantity). Instead, Boltzmann had a brilliant idea that entropy has something to do with numbers of possible microstates (or, equivalently, the probabilities of macrostates). Then, knowing that probabilities $p$ are combined multiplicatively, he realized that if $S=f(p)$, then $f(x)$ must be a logarithmic function. You see there is no "derivation" here, the key thing was the realization that entropy is related to (macro-)state probabilities. The rest comes as if by itself.