One-Dimensional Ising Model

Consider N spins in a chain which can be modeled using the one dimensional Ising model

(i)

where the spin has the values s = ±1.

a) Find the partition function.

b) Find the heat capacity per spin.

SOLUTION

a) The partition function is defined as

(1)

where the product is taken over the n sites. Define K = βJ where β = 1/τ.

Start by evaluating the sum at one end, say for n = 1. The answer is independent of the value of s₂ = ± 1:

(2)

Next we evaluate the sum over s₂, which is also independent of the value of s₃:

(3)

(4)

So each summation over s_n gives the identical factor e^K – e^-K, and Z is the product of N such factors.

b) The heat capacity per spin is obtained using thermodynamic identities. The partition function is related to the free energy F:

(5)

The entropy is given by

(6)

Now, the heat capacity C is given by

(7)

The heat capacity per spin is

(8)