Stochastic Models for the Statistical Description of Lattice Systems

Abstract

Macroscopic properties of lattice systems are described by stochastic models that permit one to construct ``typical'' configurations representing the average behavior. The transition probabilities for the models are defined with parameters expressing both short‐ and long‐range correlations; the parameters are varied to maximize a ``pseudo'' partition function and the relative worth of models can be judged in terms of these maxima. The actual construction of the lattice configurations is carried out by Monte Carlo and, since only one typical configuration is needed, quite large lattices (w≃10 5 ) can be easily described. For the simpler models the construction can be formulated analytically, in view of continuity conditions. The method was applied to the Ising problem in 2 and 3 dimensions, yielding quite accurate results with a phase transition not far off from the exact (K c ′ = 0.428 vs 0.4407) and in close agreement with series expansion (K c ′ = 0.223 vs 0.2217) , for the two cases respectively. The important point is that—in contrast to other methods—the use of stochastic models can be readily extended to more involved problems, like to the presence of external field, further neighbor interactions, or admissibility of several states for a site.