# Question about a hand-waving estimation done by Landau

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?