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.