1-dimensional Ising model

Model: Consider a system with $N$ spins in a row.

The energy of the system will be given by

$U=-J\sum _{i=1}^{N-1}S_{i}S_{i+1}$,

where each variable $S_{j}$ can be either -1 or +1.

The partition function of the system will be:

$Q_{N}=\sum _{\Omega ^{N}}\exp \left[K\sum _{i=1}^{N-1}S_{i}S_{i+1}\right]$,

where $\Omega ^{N}$ represents the possible configuration of the N spins of the system, and $K=J/k_{B}T$

$Q_{N+1}=\sum _{S_{1}}\sum _{S_{2}}e^{kS_{1}S_{2}}\sum _{S_{3}}e^{KS_{2}S_{3}}\cdots \sum _{S_{N}}e^{KS_{N-1}S_{N}}\sum _{S_{N+1}}e^{KS_{N}S_{N+1}}$

Performing the sum of the possible values of $S_{N+1}$ we get:

$Q_{N+1}=\sum _{S_{1}}\sum _{S_{2}}e^{kS_{1}S_{2}}\sum _{S_{3}}e^{KS_{2}S_{3}}\cdots \sum _{S_{N}}e^{KS_{N-1}S_{N}}\left[2\cosh(KS_{N})\right]$

Taking into account that $\cosh(K)=\cosh(-K)$

$Q_{N+1}=\sum _{S_{1}}\sum _{S_{2}}e^{kS_{1}S_{2}}\sum _{S_{3}}e^{KS_{2}S_{3}}\cdots \sum _{S_{N}}e^{KS_{N-1}S_{N}}\left[2\cosh(K)\right]$

Therefore:

$Q_{N+1}=\left(2\cosh K\right)Q_{N+1}$

$Q_{N}=2^{N}\left(\cosh K\right)^{N-1}\approx (2\cosh K)^{N}$

The Helmholtz free energy in the thermodynamic limit will be

$A=-Nk_{B}T\log \left(2\cosh K\right)$