Why Is the Multiplicity Function Considered Sharp Near Its Peak?

[P3X-018]

In order to get a "feeling" of the sharpness of the multiplicity function, for a system of 2 solids (A & B) with N quantum oscillators in each, in the high-temeperature limit (that is the total number of energy units q is much larger than N), it is approximated as a Gaussian as [Shroeder, eq. (2.27), 1999],

$$\Omega(x) = \Omega_{max}\,e^{-4Nx^2/q^2}$$

Where x comes from that if $q = q_A + q_B$ then $q_A = q/2 + x$ and $q_B = q/2 - x$. The width parameter of the above equation would be $\sigma = \frac{q}{2\sqrt{2}N}$ around x = 0 (q_A = q_B).
Now how can the function be called "sharp near it's peak", when this width parameter is such a huge number by this assumption that is q >> N?
This was used to explain that only a very small number of macrostates have a reasonable chance of occurring, but how can this be, if the width is so great?

 

[Parlyne]

Think about what $$\sigma$$ means here. It's the range about $$x=0$$ that the system will be within about two thirds of the time. So, the number you have to compare it to is just how much bigger $$x$$ could be. In other words, what percent of the total range of $$x$$ is likely.

 

[P3X-018]

But that range is very big (unless I got it completely reversed), which means, that there are many states that have a pretty high probabelity of occurring, while it should be only a VERY small fraction of the total states that could have a significant chance of occurring.
Or am I interpreting this in a wrong way?

 

[Parlyne]

Schroeder himself addresses that in the text. In reference to the width $$x = \frac{q}{2\sqrt{N}}$$, he says, "This is actually a rather large number. But if $$N = 10^{20}$$, it's only one part in ten billion of the entire scale of the graph!" And, that is the essence of the issue. What matters isn't objectively how many states have high probability. It's what fraction of the state have high probability. If we take the states within $$1\ \sigma$$ of the mean to be the likely states, then only $$\frac{q}{\sqrt{N}}$$ out of the $$q$$ possible arrangements of energy units are likely to occur. If N is large, $$\frac{q/\sqrt{N}}{q} = \frac{1}{\sqrt{N}}$$ is quite small.

 

[P3X-018]

Ahh that fraction is indeed small. Then I must've misinterpreted his argument about, that the true scale of the graf would stretch a few times around the earth. So that was a reference to, that if the "width" of the graph was as wide as shown in the picture than the tail would have to strech that far? Because it sounded like that the "width" also would stretch, but that's not the case?

 

[Parlyne]

He's saying that if the width was exactly as shown on the page, the very ends of the graph would be that far away.