Mesoscopic description of the annealed Ising model, and multiplicative noise

A new type of Langevin equation exhibiting a nontrivial phase transition associated with the presence of multiplicative noise is discussed. The equation is derived as a mesoscopic representation of the microscopic annealed Ising model ~AIM! proposed by Thorpe and Beeman, and reproduces perfectly its basic phenomenology. The AIM exhibits a nontrivial behavior as the temperature is increased, in particular it presents a disorderto-order phase transition at low temperatures, and an order-to-disorder transition at higher temperatures. This behavior resembles that of some Langevin equations with multiplicative noise, which exhibit also two analogous phase transitions as the noise amplitude is increased. By comparing the standard models for noise-induced transitions with our new Langevin equation we elucidate that the mechanisms controlling the disorder-to-order transitions in both of them are essentially different, even though for both of them the presence of multiplicative noise is a key ingredient. @S1063-651X~98!06911-6#