Monte Carlo Simulations for Small-World Stochastic Processes

by Dubreus, Terrance Maurice

We conduct a computational statistical study of nonequilibrium processes with and without small-world interactions. We first investigate the motion of a passive random walker on growing nonequilibrium one-dimensional surfaces with or without small-world connections. The walker always moves to a higher connected site on the evolving surface. The surfaces examined are related to the evolution of parallel discrete-event simulations, with or without small-world connections. We have also examined the Kim-Kosterlitz surface growth model. In particular, we study the probability distribution function of the distance between the walker and the global maximum of the surface at saturation. We find that the availability of small-world connections for either the surface or the walker dramatically changes this probability distribution function. We next report of the lifetime of the metastable state of the square-lattice Ising model. We have used a macroscopic mean-field dynamic using the density of states from a modified Wang-Landau sampling procedure. The Wang-Landau sampling, was used to give the density of states g, either as a function of two parameters, g(E,M), or as a function of only the magnetization, g(M). From the density of states the constrained free energy, F(m) , is calculated. Using a macroscopic mean-field dynamic, constrained to having only single spin flips, we obtain the lifetime, tau, of the metastable state with and without small-world connections. From F(m) we obtain the exact first-passage time, tau Comparisons to recent predictions of the droplet theory of nucleation and growth will be made.