The logarithm in the entropy formula

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Discussion Overview

The discussion revolves around the presence of the logarithm in the entropy formula, specifically the expression S = k ln(N), where k is the Boltzmann constant and N represents the number of microstates. Participants explore the implications of this formulation in terms of entropy's properties, particularly its extensiveness, and question the necessity of the logarithmic function in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the logarithm is necessary for entropy to be an extensive property, as demonstrated by the relationship between the microstates of combined systems (Post 2).
  • Others argue that the requirement for entropy to be extensive means that it should scale as S = S1 + S2, which aligns with the multiplication of microstates (Post 2).
  • A participant notes that entropy was defined as an extensive quantity prior to the development of statistical mechanics, suggesting historical context for its formulation (Post 4).
  • Another participant introduces the idea that entropy contributes to the energy of the system, indicating that this relationship also necessitates extensiveness (Post 5).
  • There is a question raised about whether the logarithm is the only function that satisfies the property f(xy) = f(x) + f(y), indicating a mathematical exploration of the function's uniqueness (Post 6, Post 8).
  • A participant discusses the derivation of entropy change in isothermal expansion and connects it to the statistical mechanics perspective, reinforcing the logarithmic relationship (Post 7).

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the necessity and implications of the logarithm in the entropy formula. There is no consensus on whether the logarithm is the only suitable function or on the broader implications of entropy's extensiveness.

Contextual Notes

Some discussions involve assumptions about the definitions of extensive and intensive properties, as well as the mathematical properties of functions. The exploration of these concepts remains unresolved.

gsingh2011
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Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?
 
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The reason that I know of, is that we require entropy to be an extensive property.
Suppose that we have two systems, with N1 and N2 microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N1N2 microstates.

However, to be an extensive quantity, the entropy should scale as
S = S1 + S2.
 
CompuChip said:
The reason that I know of, is that we require entropy to be an extensive property.
Suppose that we have two systems, with N1 and N2 microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N1N2 microstates.

However, to be an extensive quantity, the entropy should scale as
S = S1 + S2.

Why do we want entropy to be an extensive quantity? Multiplying the microstates to calculate the entropy seems just as easy/useful as adding the entropies.
 
Because Entropy was defined as an extensive quantity long before people knew about statistical mechanics.
 
Well, again there is a lot I'm omitting, but one good reason is that entropy contributes to the energy of the system as
dE = T dS - p dV + N dμ
and we definitely want that to be extensive, don't we?
(Note by the way that the quantities occur in combinations of extensive and intensive: two systems with entropy S and temperature T have total entropy 2S but temperature T, two systems with pressure p and volume V have pressure 2V but pressure p, etc)
 
But is ln the only function for which f(xy) = f(x)+f(y)?
 
delta S for n moles of a gas in isothermal expansion =
integral V1 to V2 nR dV/V = delta S= nR ln V2/V1
Given that a change in entropy in statistical mechanics from a system with probability of W1
to one of W2 = k ln W2/W1 , it should follow that
delta S = integral w1 to w2 k = k ln w2/w1
And since w2 = all the possible states in phase space and w1 = one state
Then S = k ln w
 
Last edited:
jhjensen said:
But is ln the only function for which f(xy) = f(x)+f(y)?

I answered that mathematically in your other thread
 

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