DERIVATION OF FINITE-SIZE SCALING FOR MEAN-FIELD MODELS FROM THE BURGERS-EQUATION

The magnetization per spin for a class of models with the Husimi-Temperley type interaction satisfies the Burgers equation with a diffusion coefficient 1/2N, where N is the number of particles. The role of time is played by the dimensionless interaction parameter, and the role of spatial coordinate by the dimensionless field variable. Both thermodynamic scaling and finite-size scaling for the magnetization near the critical point are derived from a family of self-similar solutions of the corresponding Burgers equation. The models are specified by the initial conditions which are chosen to correspond in the high-temperature region to a vanishing interaction constant and in the low-temperature region to an infinitely large interaction constant.