Boltzman distribution:

*Partition function* (also called normalizing constant)

Potential energy

The symbol means that they are neighboring pair, is the interaction strength, is the external magnetic field.

In zero field we can also define the potential energy as

We define the expectation of with respect to as *internal energy*:

The* free energy* of the system is

The *specific heat* of the system is

The *mean magnetization per spin* is

1.

Let , we have

Thus

2.

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