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The gaussian distribution, its properties, and its applications in statistics. The author derives sterling's approximation for large values of n and applies it to the gaussian distribution. The document also discusses the canonical representation of the gaussian distribution and its relation to the central limit theorem.
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g(N, s) = g(N, 0)e−^
(^2) Ns^2
g(N, 0) =
This assumes the following,
N 1 |s| N
P↑ = P↓ =
g(N, s) falls to (^1) e of its maximum when
s N
Goal is to find a better expression for N! when N is very large. Derivation is in the text, appendix A. We now see how we can use it.
ln N! ≈ N ln N − N +
ln (2πN ) for N 1
The error is less that 1% for N = 10. Now we apply it to g(N, 0).
ln
) 2 = ln^ N^!^ −^ 2 ln^
≈ N ln N − N +
ln (2πN ) − 2
ln
ln (πN )
Now just combine like terms. The second and fifth terms add to give 0. For the first and fourth terms,
ln N − ln
= N ln 2 = ln 2N
Now look at the terms with π
1 2
ln (2πN ) −
ln (πN )^2 =
ln
2 πN π^2 N 2
ln
πN
So the whole sum of six terms becomes
≈ ln
πN
Therefore,
g(N, 0) ≈ 2 N
πN
g(x, σ) = Ae−^
(x−¯x)^2 2 σ
where σ is the standard deviation. The area under the curve between ±σ occupies 68.3% of the total area. ± 2 σ occupies 95.4%, and ±3 occupies 99 .7%. This is a way of determining the accuracy of a given measurement. Problem is that everyone assumes all distributions are gaussian, but this may not always be the case.
Example. Find < f > if
f (s) = s^2
Using Sterling’s approximation,
g(N, s) = 2N
πN
e−^
(^2) Ns^2
< s^2 > =
s
s^2
πN
e−^ (^2) Ns^2
2 N
Now convert to an integral.
πN
−∞
s^2 e−^
(^2) Ns ds
Note that this is an approximation, since we are extending the limits of integration to infinity. If the function didn’t go to 0 at infinity and fall off quickly, we could be in trouble here. Set
x^2 =
2 s^2 N
, s =
x
2 xdx =
sds
What we need is s^2 ds. Substitute from above,
s^2 ds =
x^2
s^2
xdx √ N 2 x ︸ ︷︷ ︸ ds
=
x^2 dx
< s^2 > =
πN
−∞
x^2 e−x
2 dx ︸ ︷︷√ ︸ π 2
=
(^12)
2 · 2
1 2
< (2s^2 ) >
This is called the root mean square (rms) spin excess. Note that you can not cancel the square and the square root, to get < 2 s > (also, note that < 2 s >= 0). The Fractional rms spin excess,
< (2s)^2 >
1 2 N
Same result as before: that this is very small for large N. Means the gaussian is very sharply peaked for large N. This is why statistical mechanics works so well.
