Difference between W and Ω in Boltzmann's entropy formula

[Point Conception]

With S = K ln W where W is probability system is in the state it is in relative to all other possible states :
W = V^N , V = volume, N = number of particles so ln W = N (ln V)
And this expression is for non equilibrium state.
For equilibrium state S = K ln Ω Then is the only difference between W and Ω that Ω is for maximum entropy ?
If so it appears it would have same value as W

 

[DrClaude]

As far as a know, there is no difference in that equation between W and Ω; it is simply a different choice of symbol.

W or Ω represent the multiplicity of the corresponding macrostate. It is not a probability. (It couldn't be, because a probability ≤ 1, so S would be negative or 0, but entropy is always a positive quantity.)

Calculating entropy for a non-equilibrium system is not a trivial thing.

 

[Chandra Prayaga]

But the definition is still valid, even for a non-equilibrium state?

 

[DrClaude]

But the definition is still valid, even for a non-equilibrium state?

From https://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics

Another fundamental and very important difference is the difficulty or impossibility, in general, in defining entropy at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium; it can be done, to useful approximation, only in carefully chosen special cases, namely those that are throughout in local thermodynamic equilibrium.[1][2]

See also https://physics.stackexchange.com/questions/134377/definition-of-entropy-in-nonequilibrium-states

 

[Point Conception]

This paper by @NFuller shows a progression from S = k ln W to S = k ln Ω and how W and Ω are equated.*
https://www.physicsforums.com/insights/statistical-mechanics-part-equilibrium-systems/
So W and Ω both denote micro states however they do not seem inter changeable.
* Equated by way of Sterling's Approximation.

 

[vanhees71]

Maybe this old manuscript of mine helps, particularly the section on information theory:

https://th.physik.uni-frankfurt.de/~hees/publ/stat.pdf

 

[Point Conception]

My confusion here is that omega is used to derive S = k ln Ω :
1) S = k ln W where W = (V)^N microstates
This leads to the Gibbs statistical entropy formula for non uniform probability distribution, non equilibrium systems :
2) S = - k ∑ p_i ln p_i
Then in special case of uniform distribution p_i = 1/Ω
3) S = -k Σ 1/Ω ln 1/Ω = k ln Ω
For maximum entropy equilibrium systems
It would help to resolve omega and W if p_i = 1/W for uniform distribution

 

[Kashmir]

Does S = K ln Ω only hold in equilibrium?

 

[Point Conception]

From link in post #5