2008/01/08 by J. R. Trail · 2 citations
Mathematics · Physics and Astronomy · #Advanced Chemical Physics Studies #Applied mathematics #Central limit theorem #Computer science #Dynamic Monte Carlo method #Hybrid Monte Carlo #Limit (mathematics) #Markov chain Monte Carlo #Mathematical analysis #Mathematics #Measure (data warehouse) #Monte Carlo integration #Monte Carlo method #Monte Carlo method in statistical physics #Monte Carlo molecular modeling #Physics #Physics of Superconductivity and Magnetism #Quantum Monte Carlo #Quantum, superfluid, helium dynamics #Quasi-Monte Carlo method #Statistical physics #Statistics #cond-mat.stat-mech #physics.comp-ph #quant-ph
paper · pdf · doi:10.1103/physreve.77.016703
published as Phys. Rev. E 77, 016703 (2008) · 18 pages, 5 figures
openalex publication_date 2008/01/08 · arxiv created 2009/09/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.