Poisson Distribution in Ideal Gas

Consider a monatomic ideal gas of total molecules in a volume. Show that the probability, P_N for the number N of molecules contained in a small element of V is given by the Poisson distribution

where is the average number of molecules found in the volume V.

SOLUTION

The probability w₁ of finding a particular molecule in a volume V is

The probability w_n of finding N marked molecules in a volume V is

(1)

Similarly, the probability of finding one particular molecule outside of the volume V is

and  for particular molecules outside V,

(2)

Therefore, the probability P_N of finding any N molecules in a volume V is the product of the two probabilities (1) and (2) weighted by the number of combinations for such a configuration:

(3)

The condition  also implies that  Then we may approximate

(4)

So, (3) becomes

(5)

where we used the average number of molecules in V:

Noticing that, for large ,

we obtain

(6)

where we used

(6) can be applied to find the mean square fluctuation in an ideal gas when the fluctuations are not necessarily small (i.e., it is possible to have (N - ⟨N⟩)/ ⟨N⟩~ 1, although N is always much smaller than the total number of particles in the gas ).