Are MD (Molecular Dynamics) and MC (Monte Carlo) both parts of statistical physics or is only one of them?
If so, which branch (e.g., mechanics, thermodynamics, etc.) and why?
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Monte Carlo methods are a broad class of algorithms that apply not just to physics problems but also to biology, finance, and even pure mathematics (for example it can be used to provide better and better estimations of π, and it can be used to calculate integrals or areas under curves).
Molecular dynamics is a deterministic algorithm (meaning, it is not a Monte Carlo algorithm), however it is often done on very large ensembles of molecules, after which statistical averages are calculated (e.g. the temperature of the ensemble can be estimated by calculating the average kinetic energy over the entire ensemble of molecules, and this can be considered a branch of statistical mechanics).
As for the diagram in your question, Monte Carlo algorithms can be applied to every branch and sub-branch in the diagram, but I would put molecular dynamics in a branch called "molecular physics" which is often grouped with the branch in the top-left corner (instead of "atomic, nuclear, and particle physics", there's often a category called "atomic, molecular, and optical physics" or AMO physics, and "nuclear and particle physics" would be in a different category).