Number Fluctuations

Show that for a grand canonical ensemble the number of particles N and occupational number n_j in an ideal gas satisfy the conditions:

a)  quantum statistics

b) ⟨N²⟩ - ⟨N⟩² = ⟨N⟩ classical statistics

For an electron spin s = 1/2. Fermi gas at temperature τ << ε_F,

c) Find ⟨(∆N)²⟩.

SOLUTION

a) we have

(1)

Consider an assortment of n_k particles which are in the kth quantum state. They are statistically independent of the other particles in the gas; therefore we can apply (1) in the form

(2)

For a Fermi gas

(3)

So, by (2),

(4)

Similarly, for a Bose gas

(5)

we have

(6)

b) First solution: Since a classical ideal gas is a limiting case of both Fermi and Bose gases at ⟨n_k⟩ <<1, we get, from (3) or (6),

(7)

Alternatively, we can take the distribution function for an ideal classical gas,

and use (2) to get the same result. Since all the numbers n_k of particles in each state are statistically independent, we can write

(8)

c) Again we will use (1):

Since the gas is strongly degenerate, τ << ε_F, we can use μ = ε_F and τ = 0:

(9)

Then

(10)