Statistical Mechanics · Problem 3.32

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · two-level-system-entropy

Problem

The quantum states available to a given physical system are (i) a group of

g_1

equally likely states, with a common energy

\varepsilon_1

and (ii) a group of

g_2

equally likely states, with a common energy

\varepsilon_2 > \varepsilon_1

. Show that the entropy of the system is given by

S = -k[p_1 \ln(p_1/g_1) + p_2 \ln(p_2/g_2)],

where

p_1

and

p_2

are, respectively, the probabilities of the system being in a state belonging to group 1 or to group 2:

p_1 + p_2 = 1

. (a) Assuming that the

p_i

are given by a canonical distribution, show that

S = k\left[\ln g_1 + \ln\{1 + (g_2/g_1)e^{-x}\} + \frac{x}{1 + (g_1/g_2)e^x}\right]

where

x = (\varepsilon_2 - \varepsilon_1)/kT

, assumed positive. Compare the special case

g_1 = g_2 = 1

with that of the Fermi oscillator of the preceding problem. (b) Verify the foregoing expression for

S

by deriving it from the partition function of the system. (c) Check that at

T \to 0

S \to k \ln g_1

. Interpret this result physically.