About June 2007 I found that the mean path length in steps could scale lexical growth. It was impossible not to notice a curious aspect of the log formula, C log (n) (network entropy) that used the mean path length as a base. The mean path length was a measurable discrete length and the log function was continuous. I wondered if this discrete-continuous, duality might relate in some way to particle-wave duality in quantum mechanics. For example, if the scale factor has states each with a different integral exponent, that might represent available states. Unfortunately I don’t know much at all about quantum mechanics so I don’t know if the discrete-continuous attribute of network entropy is a meaningful coincidence or a conceptual artifact of no significance. Can you comment?
This is a very interest quest that has come into my consideration long ago when I tried to formulate my proposed new coordinate system. According to this (coordinate system) each entity is expressed by two discrete forms (duality) that could be hosted within two discrete sub-coordinate systems. The point is whether these sub-coordinate systems are both quantized or the "immaterialistic" one is continuous. I have not a conclusive answer to this (yet). However, I thing that does not make much difference in practical terms.
The so-called "entropy " doesn't exist at all.
During the process of deriving the so-called entropy, in fact, ΔQ/T can not be turned into dQ/T. That is, the so-called "entropy " doesn't exist at all.
The so-called entropy was such a concept that was derived by mistake in history.
It is well known that calculus has a definition,
any theory should follow the same principle of calculus; thermodynamics, of course, is no exception, for there's no other calculus at all, this is common sense.
Based on the definition of calculus, we know:
to the definite integral ∫T f(T)dQ, only when Q=F(T), ∫T f(T)dQ=∫T f(T)dF(T) is meaningful.
As long as Q is not a single-valued function of T, namely, Q=F( T, X, …), then,
∫T f(T)dQ=∫T f(T)dF(T, X, …) is meaningless.
1) Now, on the one hand, we all know that Q is not a single-valued function of T, this alone is enough to determine that the definite integral ∫T f(T)dQ=∫T 1/TdQ is meaningless.
2) On the other hand, In fact, Q=f(P, V, T), then
∫T 1/TdQ = ∫T 1/Tdf(T, V, P)= ∫T dF(T, V, P) is certainly meaningless. ( in ∫T , T is subscript ).
We know that dQ/T is used for the definite integral ∫T 1/TdQ, while ∫T 1/TdQ is meaningless, so, ΔQ/T can not be turned into dQ/T at all.
that is, the so-called "entropy " doesn't exist at all.
Why did the wrong "entropy" appear ?
In summary , this was due to the following two reasons:
1) Physically, people didn't know Q=f(P, V, T).
2) Mathematically, people didn't know AΔB couldn‘t become AdB directely .
If people knew any one of them, the mistake of entropy would not happen in history.
Please read my paper and those answers of the questions related to my paper in my Projects.
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity
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