Statistical Mechanics · Problem 4.3

Formulation of Quantum Statistics — R. K. Pathria

Physics — Statistical Mechanics · particle-number-fluctuations-grand-canonical

Problem

A vessel of volume

V^{(0)}

contains

N^{(0)}

molecules. Assuming that there is no correlation whatsoever between the locations of the various molecules, calculate the probability,

p(N, V)

, that a region of volume

V

(located anywhere in the vessel) contains exactly

N

molecules. ((i)) Show that

\bar{N} = N^{(0)}p

and

(\Delta N)_{\text{r.m.s.}} = \{N^{(0)}p(1-p)\}^{1/2}

, where

p = V/V^{(0)}

. ((ii)) Show that if both

N^{(0)}p^{(0)}

and

N(1-p)

are large numbers, the function

p(N, V)

assumes a Gaussian form. ((iii)) Further, if

p \ll 1

and

N \gg N^{(0)}

, show that the function

p(N, V)

assumes the form of a Poisson distribution:

p(N) = e^{-\bar{N}} \frac{(\bar{N})^N}{N!}.