Editing 1-dimensional Ising model
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Model: | |||
Consider a system with <math> N </math> spins in a row. | |||
The energy of the system will be given by | |||
<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>, | |||
where each variable <math> S_j </math> can be either -1 or +1. | where each variable <math> S_j </math> can be either -1 or +1. | ||
The | The partition function of the system will be: | ||
<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>, | |||
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and <math> K = J/k_B T </math> | and <math> K = J/k_B T </math> | ||
<math> 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} } | |||
</math> | </math> | ||
Performing the sum of the possible values of <math> S_{N} </math> we get: | Performing the sum of the possible values of <math> S_{N} </math> we get: | ||
<math> 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-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right] | |||
</math> | </math> | ||
Taking into account that <math> \cosh(K) = \cosh(-K) </math> | Taking into account that <math> \cosh(K) = \cosh(-K) </math> | ||
<math> 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] | |||
</math> | </math> | ||
Therefore: | Therefore: | ||
<math> Q_{N} = \left( 2 \cosh K \right) Q_{N-1} </math> | |||
<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math> | |||
The | The Helmholtz free energy in the thermodynamic limit will be | ||
<math> A = - N k_B T \log \left( 2 \cosh K \right) </math> | |||