Absolute Entropy (Statistical)

[Master J]

I was looking at a derivation of entropy expressed as an absolute probability:

S = -k. SUM P.lnP
(What is the name of this by the way?)

In the derivation, it makes the following statements which I really just don't get!

U = SUM E.P

so therefore dU = SUM E.dP - SUM P.dE

Where does the minus sign come from? Should it not be a plus??

Then, it goes from dS = -k. SUM lnP.dP to dS = -k.d(SUM P.lnP)

How is that true?? First it was just P that was a differential element, now its the whole expression in the bracket??

 

[Gerenuk]

How does the whole derivation go?

U = SUM E.P
so therefore dU = SUM E.dP - SUM P.dE
Where does the minus sign come from? Should it not be a plus??

I agree.

Then, it goes from dS = -k. SUM lnP.dP to dS = -k.d(SUM P.lnP)

How is that true?? First it was just P that was a differential element, now its the whole expression in the bracket??

Check what the whole expression is equal to. By the product rule:
$$\mathrm{d}(\sum P_i\ln P_i)=\sum(\ln P_i\mathrm{d}P_i+P_i\mathrm{d}\ln P_i)=\sum\ln P_i\mathrm{d}P_i+\sum P_i\frac{\mathrm{d}P_i}{P_i}=\sum\ln P_i\mathrm{d}P_i+\sum\mathrm{d}P_i$$
The last term is zero since
$$\sum P_i=1$$
and hence
$$\sum \mathrm{d}P_i=\mathrm{d}(\sum P_i)=\mathrm{d}1=0$$

 

[astrokreng]

The derivation you are referring to is known as the Boltzmann entropy, which is a statistical definition of entropy based on the probability of a system being in a particular state. This approach is often used in thermodynamics and statistical mechanics to describe the behavior of systems at the microscopic level.

Regarding your question about the minus sign in the expression dU = SUM E.dP - SUM P.dE, this comes from the fact that energy (E) and probability (P) have opposite signs. When we take the derivative of U with respect to P, we need to take into account this sign difference, which results in the minus sign in front of the second term.

As for your confusion about the change from dS = -k. SUM lnP.dP to dS = -k.d(SUM P.lnP), this is simply a mathematical manipulation known as the chain rule. In the first expression, we are taking the derivative of lnP with respect to P, which results in dP in the denominator. In the second expression, we are taking the derivative of the entire expression SUM P.lnP, which results in d(SUM P.lnP) in the denominator. The two expressions are equivalent and both are used in different contexts.

I hope this helps clarify your questions about the derivation of absolute entropy. It is a complex concept, but it is a fundamental aspect of understanding the behavior of systems at the microscopic level.