Help Understanding Gaussian Particle Distribution

Question

I have a simple enough problem. I want to mathematically describe a Gaussian particle distribution of $N$ total particles with a spot size of 0.1 (sigma). I need a function that will tell me the amount of particles in a transverse position $x$, i.e. how many particles there are in a bin kind of. But, the distribution should meet three physical requirements:

1. When integrated in all space (from -Infinity to +Infinity) the result should be N.

2. The spot size of the distribution is sigma (0.1, for example).

3. The values of $N$ at any given $x$ should not exceed the total number of particles $N$.

Now, of course the standard equation for a Gaussian is:

$$ \ N(x) = \frac{N} {\sigma\sqrt{2\pi}} e^{-x^2/(2\sigma^2)} \,. $$

But, this satisfies condition 1 and 2 but not 3! I need an equation that describes the distribution in real space (meaning that for any $x$ value, it should give the number of particles in that transverse slice) but I am confused on how I to go about doing that.

Or is even possible to meet all three conditions with a Gaussian distribution? Any help would be greatly appreciated!

Answer 1

The problem is that $N(x)/N$ is a probability density, not a probability. You can see that from the units: $N/\sigma$ has units "number per meter". You have to ask: How many balls will I find between $x_1$ and $x_2$?

The expected number of balls is calculated from this density: $$ \bar{N} \approx N(x) \Delta x$$ if $\Delta x = x_2-x_1$ is small enough. If it's too large, then you have to integrate: $$\bar{N} = \int_{x_1}^{x_2} N(x)\,\mathrm{d}x$$

$\bar{N}$ will always be less than $N$.