When we combined A and B we said that we need to keep them separated, so the total multiplicity is the product of the multiplicities, instead of having a system with 20 particles and 40 cells, which would give a much higher total multiplicity. Does this mean we need to be careful when we divide a system into 2 and say that the total entropy is the sum of the entropies? If I conceptually divide (with an imaginary boundary) a box containing a gas into 2 equal parts, I cannot say the entropy of a part is half the total entropy. That would be true as long as particles cannot go from one side to the other, which is allowed in the original system. For other extensive properties like the number of moles I wouldn't have this constraint and additivity works. This is very tricky.
Yes, well spotted. To make this more rigorous, we would need to assume that V and N are both ≫ 1. In that case, the (fractional) difference between ln C(2V,2N) and 2 ln C(V,N) becomes arbitrarily small. For the small numbers in this example, they are not the same. But luckily, most macroscopic thermodynamic systems contain enough molecules that this is not a concern. For macroscopically large systems, the entropy of half of the system is indeed (very, very close to) half of the total entropy. This is what we are working our way towards proving.
This video describes the connection between Boltzmann's formula (S = k ln W) and the Gibbs entropy (S = −k Σᵢ pᵢ ln pᵢ): th-cam.com/video/7V-IHpiTk2I/w-d-xo.html
I find your tutorials always concise, it has helped me a lot in understanding thermodynamics.
I couldn't hope to hear anything better, thanks!
Your last line is just OP much much accurate & meaningful definition of Entropy .
Glad you stuck around until the end 😊
Top notch that it is simple and straight forwardly explained.
When we combined A and B we said that we need to keep them separated, so the total multiplicity is the product of the multiplicities, instead of having a system with 20 particles and 40 cells, which would give a much higher total multiplicity. Does this mean we need to be careful when we divide a system into 2 and say that the total entropy is the sum of the entropies? If I conceptually divide (with an imaginary boundary) a box containing a gas into 2 equal parts, I cannot say the entropy of a part is half the total entropy. That would be true as long as particles cannot go from one side to the other, which is allowed in the original system. For other extensive properties like the number of moles I wouldn't have this constraint and additivity works. This is very tricky.
Yes, well spotted.
To make this more rigorous, we would need to assume that V and N are both ≫ 1. In that case, the (fractional) difference between ln C(2V,2N) and 2 ln C(V,N) becomes arbitrarily small. For the small numbers in this example, they are not the same. But luckily, most macroscopic thermodynamic systems contain enough molecules that this is not a concern.
For macroscopically large systems, the entropy of half of the system is indeed (very, very close to) half of the total entropy. This is what we are working our way towards proving.
what a beautifull attitude.. you deserve your title. thanks i learn new attitudes from you
That's very nice of you, thanks.
Can you make a video on Gibbs Entropy too,(while also talking about what makes it different from Boltzmann entropy.)
This video describes the connection between Boltzmann's formula (S = k ln W) and the Gibbs entropy (S = −k Σᵢ pᵢ ln pᵢ):
th-cam.com/video/7V-IHpiTk2I/w-d-xo.html