Statistical Mechanics · Problem 8.12

Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions — R. K. Pathria

Physics — Statistical Mechanics · fermi-bose-specific-heat-two-dimensions

Problem

Show that, in two dimensions, the specific heat

C_V(N, T)

of an ideal Fermi gas is identical with the specific heat of an ideal Bose gas, for all

N

and

T

. [Hint: It will suffice to show that, for given

N

and

T

, the thermal energies of the two systems differ at most by a constant. For this, first show that the fugacities,

z_F

and

z_B

, of the two systems are mutually related:

(1 + z_F)(1 - z_B) = 1, \quad \text{i.e.} \quad z_B = z_F/(1 + z_F).

Next, show that the functions

f_2(z_F)

and

g_2(z_B)

are also related:

f_2(z_F) = \int_0^{z_F} \frac{\ln(1+z)}{z}\,dz = g_2\left(\frac{z_F}{1+z_F}\right) + \frac{1}{2}\ln^2(1+z_F).

It is now straightforward to show that

E_F(N,T) = E_B(N,T) + \text{const.},

the constant being

E_F(N,0)