Statistical physics, as a branch of modern physics, uses methods of probability theory and statistics to solve physical problems with large populations and approximations. In this thesis, we use numerical simulations to study two statistical physical models — Ising model under topological constraints and particle systems with anomalous behavior.Ising model is a mathematical model of ferromagnetism, which describes how magnetic spins, with values -1 or 1, change their states under nearest neighbor interactions and the external magnetic field. We study a topologically constrained Ising model, where several pre-selected anchored sites are fixed to be value 1, and the topology of the active domain (the union of all value 1 sites) remains invariant under the evolution of the system. When the sites change their values with less preference of 1, the system tends to an equilibrium that approximates the Steiner tree structure. For two- to four-anchor cases, we calculate the theoretical equilibrium configurations, and in particular for three and four anchors, the positions of the Steiner points. For one-anchor case, we consider a reversed model that a single active site grows to a coral-shape active domain. In all analysis, we provide simulation results for verification.The second part of the thesis is devoted to study particle system with anomalous behavior. Anomalous behavior originates from the Braess Paradox, which states that adding an extra path to a network could in some cases impede the overall performance. We study and reproduce a spring-string model by Cohen and Horowitz in mechanical network exhibiting such paradoxical behavior. We simulate their model in two different ways and in both ways the anomalous behavior is observed. We also identify the conditions of the system parameters for the anomalous behavior and verify our theoretical results via simulations.
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