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1-dimensional Ising model

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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[edit]

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