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===Ising's exact solution===
In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The Hamiltonian of the one-dimensional Ising model on a lattice of ''L'' sites with
<math display="block">H(\sigma) = -J \sum_{i=1,\ldots,L-1} \sigma_i \sigma_{i+1} - h \sum_i \sigma_i,</math>
where ''J'' and ''h'' can be any number, since in this simplified case ''J'' is a constant representing the interaction strength between the nearest neighbors and ''h'' is the constant external magnetic field applied to lattice sites. Then the
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