2、Problem of Planck’s entropy theory for long-distance interaction system

Most contribution of Planck was on quantum physics. But he had his own view in thermodynamics. And it became part of classical thermodynamics.
The following is deduction of probability in thermodynamics. Planck found that there is monotonic change between entropy of independent system and probability in thermodynamics. He guessed that there is some relation as the following between the entropy and the probability.
S=f（W）
Suppose there are two independent parts in this system,
S------- Entropy of the system
S1----- Entropy of part 1
S2------- Entropy of part 2
W------- Overall Probability
W1----- Probability of part 1
W2----- Probability of part 2
Suppose S=S1+S2=f（W）
S1=f（W1）
S2=f（W2）
W=W1*W2
Through infinitesemal calculus, we get
S=k*In（W）----------（2）
See “Statistical Physics Introduction” the second edition by Zhuxi Wang

Probability relation W=W1*W2 requires the two parts are independent. Can all subjects meet the requirement? Actually there is something that breaks their independency. For example, there are two cups of water on a table. (which can be viewed as two parts of the system, the segmentation is random) there are a lot of charges floating there, so there is long-distance interaction, so independency is meaningless here. There is shortage in Planck deduction.
S=k*In（W）is an important core formula in thermodynamics. That there is some mistake in its deduction means there should be some adjustment for the original thermodynamics.