The Monte Carlo method is a technique to sample configurations of the state of a system governed by stochastic dynamics. The traditional Metropolis method based on a Markovian random-walk that forms the basis of the Monte Carlo method, is regularly applied to low-dimensional Hamiltonians describing strongly correlated materials such as superconductors and correlated magnets to map their temperature – electron-doping phase-diagrams leading to valuable insights, such as the physics underlying high-Tc superconductors. Combined with the Schrodinger Equation, the metropolis method gives rise to Quantum Monte-Carlo (QMC), which is the most accurate method we know of to predict ground-state properties of both solids and molecules. Non-Markovian random walks based Monte-Carlo methods, such as the Wang-Landau method is ideally suited to study problems where local interactions are much stronger with strong frustration in the dynamic variables, and where sampling is done with a constant probability making it faster than the Metropolis method. This technique has been applied to study protein folding; multiple-scattering physics due to local magnetic defects in solids, and classical Heisenberg spin systems with great success. Combining Monte-Carlo methods with Molecular Dynamics, especially ab Initio Molecular Dynamics (AIMD), can be a powerful scheme to obtain thermodynamic properties of non-ergodic or glassy systems while capturing key quantum mechanical effects including reactions.
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