Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 203

4.83 One-Dimensional Ising Model (Tennessee)

a) The partition function is defined as

where the product is taken over the sites. Define where
Start by evaluating the sum at one end, say for The answer is
independent of the value of

Next we evaluate the sum over which is also independent of the value
of

So each summationover gives the identical factor 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:

The entropy is given by

Now, the heat capacity C is given by

The heat capacity per spin is