Monte Carlo Methods: a powerful tool of statistical physics
Statistical mechanics of condensed matter systems (solids, fluids) tries to express macroscopic equilibrium properties of matter as averages computed from a Hamiltonian that expresses interactions of an atomistic many body system. While analytic methods for most problems involve crude and uncontrolled approximations, the Monte Carlo computer simulation method allows a numerically exact treatment of this problem, apart from “statistical errors” which can be made as small as desired, and the systematic problem that a system of finite size is treated rather than the thermodynamic limit. However, the simulations of phase transitions then elucidate how a symmetry breaking arises via breaking of ergodicity, if the Monte Carlo sampling is interpreted as a time average along a stochastic trajectory in phase space, in the thermodynamic limit. These concepts are illustrated for the transition paramagnet-ferromagnet of the Ising model, and unmixing transitions in polymer mixtures. As an example of the application of Monte Carlo to clarify questions about dynamic processes, simulations of interdiffusion in lattice models of alloys are discussed.
KeywordsMonte Carlo Ising Model Thermodynamic Limit Finite Size Monte Carlo Step
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