Final remark on equivalence in statistical mechanics

The analysis that we have just carried out has shown an equivalence between physical models. In statistical mechanics, the term equivalence can be used with two different "nuances":

  • exact equivalence: this means that there is an exact mapping between the partition functions of two different models
  • approximate equivalence: this means that the partition functions of two models are not related by an exact mapping, but nevertheless they behave in exactly the same way near a critical point. Such models are said to belong to the same universality class.

Apart from the equivalence between the Ising model and neural networks, all the correspondences we have shown are exact.