### Markov-chain Monte Carlo mothod (MCMC)

Efficient way of computing statistical average of physical quantities at equilibrium by replacing the statistical summation over all microscopic states by stochastic sampling. It is often called just “Monte Carlo method”. For example, in the case of Ising model, the total number of microscopic states increases as a function of the number of spins. Therefore, it is practically impossible to computer the expectation values strictly following the definition. In Markov-chain Monte Carlo method, a stochastic process is defined so that it satisfies the ergodic condition and the balance condition. Temporal averages over the microscopic states generated in this way should equal the thermal averages at equilibrium. Slow relaxation is often problematic for systems near the criticality or with frustration. There are a number of techniques designed for dealing with this problem, such as extended ensemble methods and variational Monte Carlo method.

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### Molecular Dynamics (MD)

Methods of simulating many particle systems by solving equations of motion such as the Newton equation. Mathematically, in molecular dynamics simulation one solves a system of simultaneous ordinary differential equations by using, e.g., the Runge-Kutta method and the velocity Verlet method. While a simple implementation would allow simulation with fixed energy and fixed volume, introducing the Nose-Hoover thermostat makes it possible to simulate with fixed temperature possible as well. Similarly, simulation with fixed pressure or with chemical potential is possible. There are various choices of the force field, the interaction energy between particles, ranging from the simple short-ranged force such as the hard sphere potential and the Lennard-Jones potential to the long-ranged one such as the Coulomb potential or more realistic and more complicated ones, depending on the purpose of the simulation.

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### Monte Carlo

A method of simulation is called a Monte Carlo method if sampling with pseudo random numbers is used. The simplest example is the random sampling with the weight that is uniform in the configuration space. An important category is that of importance sampling methods, e.g., Markov-chain Monte Carlo. The method is also used for solving optimization problems via simulated annealing.

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