The Fundamental Postulate says:
In equilibrium, all accessible microstates are equally likely.
Accessible means having same energy.(right?)
Let a container is taken full of gas having number of particles $N_,$ volume $V$ and energy $E\:_;$ the system is isolated.
At equilibrium, the system would be in that macrostate which would have the maximum multiplicity or the largest number of microstates; that would correspond to gas totally dispersed over the whole volume $V\;.$
However, consider the case when the gas gets confined to the left half of the container that is in volume $V/2\;_;$ this macrostate would have a much lesser multiplicity than the former one that is $$\Omega(N,V,E)\gt \Omega\left(N,\frac{V}{2}, E\right) $$
However, as the microstates corresponding to $(N,V/2,E)$ have the same energy $E\;_,$ all the microstates of $(N,V,E)$ and $(N,V/2,E)$ are equally likely in equilibrium.
But, since, the microstate corresponding to $N,V/2,E$ doesn't represent the microstate corresponding to the maximum entropic macrostate, all the microstates $\Omega\left(N,\frac{V}{2}, E\right)$ must be called fluctuation.
But fluctuation, in equilibrium?
At first, I couldn't believe this; how fluctuation happens in equilibrium.
But according to the Fundamental Postulate, all microstates having the same energy are equally likely in equilibrium. This means the microstates of the fluctuation can be exhibited by the system in equilibrium as they are equally likely along the other microstates having the same energy $E.$
How can this happen? How can fluctuation microstates occur in equilibrium, inspite of having the same energy $E\,?$
I thought I was mistaking; only the microstates corresponding to $(N,V,E)$ can occur in equilibrium. But also, I can't deny the fact the the microstates having the same energy $E$ are equally likely.
Also, as written here:
We will first consider an isolated system, typically a gas enclosed in a box, which is thermally insulated. So, any time evolution of the system will be subject to the constraint that the total energy remains constant. Left for a long time, it is believed to be in equilibrium. We further assume that given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates. This is the postulate of equal a priory probability. ... Now each macrostate comprises of numerous microstates. For example, all the gas confined to only one half of the box, is a macrostate. There is a huge number of ways this can happen, by various arrangements of particles and their momenta. The gas uniformly occupying the whole volume of the box, is another macrostate. And again, there are a huge lot of microstates associated with this macrostate. Now each microstate is equally probable, but we never actually see a gas occupying only one half of its container. Why does that happen? It happens because the number of microstates associated with the gas occupying the whole volume are overwhelmingly large, compared to the microstates associated with the gas occupying only one half of the box.
Notice the words, 'equilibrium', 'each' and the follow-up question 'Why does that happen'?; the author clearly means that at equilibrium, since all the microstates are equally likely, not only the macrostate having the gas uniformly spread over the entire volume has the greatest probability to appear; but also the macrostate corresponding to the gas confined to the left-half of the container can appear in equilibrium; it is only that the former is exhibited mostly.
So, as the Fundamental Postulate permits all the microstates having the same energy $E$ to occur equally likely in equilibrium, it is inferred from that the fluctuation occurs in equilibrium since the microstates $\Omega(N,V/2,E)$ are equally likely in equilibrium :(
So, my question is:
- Does fluctuation occur really in equilibrium as its microstates are permitted to occur equally likely in the system at equilibrium? Or is it wrong?
What am I actually missing in utilising the Fundamental Postulate and fluctuation? Can anyone please help me clear out my confusion?