We study the order parameter probability distribution at the critical point for the three-dimensional spin-1/2 and spin-1 Ising models on the simple cubic lattice under periodic boundary conditions. The finite size scaling relation for the order parameter probability distribution is tested and verified numerically by microcanonical Creutz cellular automata simulations. The state critical exponent δ, which characterizes the far tail regime of the scaling order parameter probability distribution, is estimated for three-dimensional Ising models using the cellular automaton simulations at the critical temperature. The results are in good agreement with the Monte Carlo calculations.
The single and joint distribution functions of the nearest-neighbor cell magnetization of the three-dimensional Ising model have been studied by Monte Carlo methods. Near the critical point, both distribution functions tend, for large
L, towards scaled universal forms. Estimates of the order parameter and susceptibility are obtained from the distribution functions. The critical temperature is estimated by a method independent of the estimate of the critical exponents. The critical exponent $\ensuremath{\nu}$ is obtained with the use of finite-size scaling arguments. The joint distribution function shows a double-well structure for
T<Tc and can be represented by a coarse-grained effective Hamiltonian of two cell variables. We have determined the dependence of this functional on the coarse-graining cell size for several temperatures. In particular, we have calculated the dependence of the "spinodal curve" on the (scaled) coarse-graining size. The relevance of this work to the kinetics of first-order phase transitions is also discussed.
We study the probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-state generalization known to have remarkably small corrections to scaling. Both models are studied in a cubic box with periodic boundary conditions. The model with reduced corrections to scaling makes it possible to determine P(M) with unprecedented precision. We also obtain a simple, but remarkably accurate, approximate formula describing the universal shape of P(M).