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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{N+1}} we get:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 $\cosh(K)=\cosh(-K)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{N+1} = \left( 2 \cosh K \right) Q_{N} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = - N k_B T \log \left( 2 \cosh K \right) }