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# 1-dimensional Ising model

The 1-dimensional Ising model is an Ising model that consists of a system with $N$ spins in a row. The energy of the system is 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 $K = J/k_B T$

$Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}}\sum_{S_{N}} e^{K S_{N-1} S_{N} }$

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

$Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]$

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

$Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K ) \right]$

Therefore:

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

The Helmholtz energy function in the thermodynamic limit will be

$A = - N k_B T \log \left( 2 \cosh K \right)$

## References

1. Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 2 (freely available pdf)