Metropolis Monte Carlo in Physics

The Metropolis algorithm is the workhorse of statistical-physics simulation: by accepting or rejecting proposed moves based on their energy cost, it builds a Markov chain that samples configurations with their correct Boltzmann probability.

Definition

The Metropolis algorithm is a Markov chain Monte Carlo method that generates a sequence of configurations whose limiting distribution is the canonical ensemble, by proposing local changes and accepting them with a probability set by the Boltzmann factor of the energy change.

Scope

This topic covers the Metropolis and Metropolis-Hastings algorithms as applied to physical systems: the acceptance rule, detailed balance and ergodicity, equilibration and autocorrelation, and the estimation of thermal averages and their statistical errors. It is the foundational sampling method underlying the broader Monte Carlo area.

Core questions

• How does the acceptance probability depend on the energy change of a proposed move?
• Why does detailed balance guarantee the correct stationary distribution?
• How are equilibration and autocorrelation times diagnosed and accounted for?
• How is the statistical error of a Monte Carlo average estimated from correlated samples?

Key theories

Detailed balance and stationarity

Choosing acceptance probabilities that satisfy detailed balance with respect to the Boltzmann distribution ensures that distribution is stationary under the Markov chain, so long-run averages converge to thermal expectation values.

Metropolis-Hastings generalization

Hastings generalized the acceptance rule to asymmetric proposal distributions, broadening the algorithm beyond symmetric local moves while preserving the target stationary distribution.

Autocorrelation and error estimation

Successive Metropolis samples are correlated, so the effective number of independent samples is reduced by the autocorrelation time, which must be measured to assign honest error bars to thermal averages.

Clinical relevance

Metropolis sampling computes thermodynamic quantities of lattice spin models, fluids and polymers, locates phase transitions, and serves as the core engine within Monte Carlo molecular simulation and many quantum Monte Carlo schemes.

History

Introduced in 1953 to compute the equation of state of a two-dimensional hard-disk fluid on the MANIAC computer at Los Alamos, the algorithm was generalized by Hastings in 1970 and became the most widely used simulation method in statistical physics and, later, in Bayesian statistics.

Key figures

• Nicholas Metropolis
• Arianna Rosenbluth
• W. Keith Hastings

Seminal works

• metropolis1953
• hastings1970

Frequently asked questions

Why are moves that lower the energy always accepted?

A move that lowers the energy increases the Boltzmann weight, so accepting it always moves the chain toward more probable states; uphill moves are accepted only sometimes, with a probability set by the energy increase, which is what lets the chain explore the full thermal distribution rather than rolling only downhill.

Why must samples be discarded at the start of a run?

The chain starts from an arbitrary configuration that is not yet representative of the equilibrium distribution. The initial equilibration or burn-in period is discarded so that the measured averages reflect the true thermal ensemble rather than the starting bias.