Two-Level System

Consider a system composed of a very large number N of distinguishable atoms at rest and mutually noninteracting, each of which has only two (nondegenerate) energy levels: 0, ε > 0. Let E / N be the mean energy per atom in the limit N → ∞.

a) What is the maximum possible value of E / N if the system is not necessarily in thermodynamic equilibrium? What is the maximum attainable value of E / N if the system is in equilibrium (at positive temperature)?

b) For thermodynamic equilibrium compute the entropy per atom S/N as a function of E / N.

SOLUTION

a) There is nothing to prevent giving each atom its larger energy ε, hence, η = E/Nε has a maximum of 1 with E/N = ε. Clearly, the system would not be in thermal equilibrium. To compute the problem in equilibrium, we need to determine the partition function, Z. For distinguishable noninteracting particles, the partition function factors, so for identical energy spectra

(1)

The free energy would be

(2)

The energy is then

(3)

or

(4)

where x = e^-ε/τ. Obviously, since both ε and τ are positive, x cannot be larger than 1. On the other hand, x/(1 + x) is a monotonic function which goes to 1/2 when τ goes to infinity; hence, max{E/Nε} = f(1) = 1/2 at τ → ∞.

b) The entropy may be found from (2)–( 4):

(5)

The entropy per particle, s = S/N, is given by

(6)

Writing

(7)

We can check that

(8)

as it should.