Mean-field theory of the general-spin Ising model

Motivated by modelling in physics and other disciplines, such as sociology and psychology, we derive the mean field of the general-spin Ising model from the variational principle of the Gibbs free energy. The general-spin Ising model has $$2k+1$$ 2 k + 1 spin values, generated by $$-(k-j)/k$$ - ( k - j ) / k , with $$j=0,1,2\ldots ,2k$$ j = 0 , 1 , 2 … , 2 k , such that for $$k=1$$ k = 1 we obtain $$-1,0,1$$ - 1 , 0 , 1 , for example; the Hamiltonian is identical to that of the standard Ising model. The general-spin Ising model exhibits spontaneous magnetisation, similar to the standard Ising model, but with the location translated by a factor depending on the number of categories $$2k+1$$ 2 k + 1 . We also show how the accuracy of the mean field depends on both the number of nodes and node degree, and that the hysteresis effect decreases and saturates with the number of categories $$2k+1$$ 2 k + 1 . Monte Carlo simulations confirm the theoretical results. Graphic abstract