A toy model for localized to itinerant electron transitions

A toy statistical model which mimics localized-to-itinerant electron transitions is introduced. We consider a system of two types of charge carriers: (1) localized ones, and (2) itinerant ones. There is chemical equilibrium between these two. The solution of the model shows that there is a crossover, at a specific value of a local interaction energy parameter $J$, at which carriers transfer from being itinerant to a localized/mixed valence state. The model has a very crude analogy to the observed $\gamma\rta\alpha$ transition in metal cerium.