When the system is a physical system in contact with a heat bath, each state r is weighted according to the Boltzmann factor, P r ( r i ) ∝ ∝ exp ⁡ ⁡ ( − − β β F r ) . In Monte Carlo, the canonical ensemble is defined by choosing π π ( r ) to be proportional to P r ( r i ) . In this situation, the estimator corresponds to a simple arithmetic average: