Metropolis with noise: The penalty method

The Metropolis method, when applied to the Boltzmann distribution, uses the energy difference between two states of the system in the acceptance probability. If that energy difference is a statistical estimate, rather than an exact value, the output of the Metropolis algorithm will be biased. If the noise is normally distributed, the Metropolis algorithm can be corrected by modifying the form of the acceptance probability. We call this the penalty method because the correction causes additional rejections. One application of this technique uses Quantum Monte Carlo (QMC) to compute interatomic potentials during each step of a classical Monte Carlo simulation. The energies from the QMC calculation are noisy, and the penalty method corrects the sampling of the classical MC simulation. We apply this to fluid molecular hydrogen.