I was reading the book written by Landau on statistical mechanics (Statistical Physics I 3rd Ed). In §1 there is a footnote that Landau wants to illustrate how accurate a statistical average can be for a macroscopic body.

We may give an example to illustrate the very high degree of accuracy with which this is true. If we consider a region in a gas which contains, say, 1/100 gram-molecule, we find that the mean relative variation of the energy of this quantity of matter from its mean value is only $\sim 10^{-11}$. The probability of finding (in a single observation) a relative deviation of the order of $10^{-6}$, say, is given by a fantastically small number, $\sim 10^{-3 \times 10^{15}}$.

Indeed I could get the first number $10^{-11}$ but not the second $10^{-3 \times 10^{15}}$. The way I did it is as follows:

For simplicity, consider 1/100 grams of He gas molecules (1/400 mol), which is almost ideal at room temperature. The relative deviation of energy is given by

$\dfrac{\Delta E}{\bar{E}} \sim \dfrac{1}{\sqrt{N}} = 3 \times 10^{-11}$

I then use the normal distribution PDF to estimate the second number:

$f(E|\bar{E},\Delta{E}) \sim \exp \Big[ - \dfrac{1}{2} \Big(\dfrac{E - \bar{E}}{\Delta E} \Big)^2 \Big] = \exp \Big[ - \dfrac{1}{2} \Big( \dfrac{10^{-6} \bar{E}}{\Delta E} \Big)^2 \Big] \sim 10^{- 10^9}$

and the difference in the order of magnitude is huge. Can anyone help what has just happened?


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