Fluctuating local field method probed for a description of small classical correlated lattices

Thermal-equilibrated finite classical lattices are considered as a minimal model of the systems showing an interplay between low energy collective fluctuations and single site degrees of freedom. Standard local field approach, as well as classical limit of the bosonic DMFT method, do not provide a satisfactory description of Ising and Heisenberg small lattices subjected to an external polarizing field. We show that a dramatic improvement can be achieved within a simple approach, in which the local field appears to be a fluctuating quantity related to the low energy degree(s) of freedom.