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:

$Q_N={\sum }_{{\mathrm{\Omega }}^N}\exp [K\sum _{i=1}^{N-1}{S}_i{S}_{i+1}]$,

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

$Q_N={\sum }_{S_1}{\sum }_{S_2}{e}^{K{S}_1{S}_2}{\sum }_{S_3}{e}^{K{S}_2{S}_3}\cdots {\sum }_{S_{N-1}}{e}^{K{S}_{N-2}{S}_{N-1}}{\sum }_{S_N}{e}^{K{S}_{N-1}{S}_N}$

Performing the sum of the possible values of $S_N$ we get:

$Q_N={\sum }_{S_1}{\sum }_{S_2}{e}^{K{S}_1{S}_2}{\sum }_{S_3}{e}^{K{S}_2{S}_3}\cdots {\sum }_{S_{N-2}}{e}^{K{S}_{N-2}{S}_{N-1}}[2\cosh (K{S}_{N-1})]$

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

$Q_N={\sum }_{S_1}{\sum }_{S_2}{e}^{K{S}_1{S}_2}{\sum }_{S_3}{e}^{K{S}_2{S}_3}\cdots {\sum }_{S_{N-1}}{e}^{K{S}_{N-2}{S}_{N-1}}[2\cosh (K)]$

Therefore:

$Q_N=(2\cosh K)Q_{N-1}$

$Q_N=2^N{(\cosh K)}^{N-1}\approx (2\cosh K{)}^N$

The Helmholtz free energy in the thermodynamic limit will be

$A=-N{k}_{B}T\log (2\cosh K)$