Question

I'm trying to derive a mean field expression of the dependence of the mean field of a 1D Ising ring model with σᵢ = 0,1.

What I have derived so far is:

The Hamiltonian

H = −J ∑ᵢ σᵢ σᵢ + 1

The partition function

Z=Πᵢ ∑ σᵢ = 0,1exp ( −J / k_bT ∑ᵢ σᵢσᵢ + 1 )

Taking account all the possible products:

∑σᵢ=0,1 σᵢ σᵢ + 1 = 1 × 1 + 1 × 0 + 0 × 1 + 0 × 0 = 1

Then we have
Z = Πᵢ ( e^N J / k_bT + 3 )^N

This is here where I'm stuck.

I eventually want to add a weak magnetic field see how the system behaves as a function of J.

Answer 1

To derive the mean field expression for the 1D Ising ring model, you need to calculate the partition function. Correctly calculating the partition function will give you the mean field expression. To include a weak magnetic field, modify the Hamiltonian by adding a term for the interaction of the spins with the magnetic field.

Step-by-step explanation:

To derive the mean field expression for the 1D Ising ring model, you need to calculate the partition function. However, your current expression for the partition function is incorrect. The correct expression is Z = Πi (eJN/kBT + 3)^N, where N is the number of spins in the ring.

To include a weak magnetic field, you would need to modify the Hamiltonian by adding a term for the interaction of the spins with the magnetic field. This additional term will affect the behavior of the system as a function of J.