The resolution of Monte Carlo (MC) computer simulations has steadily increased as computers have become more powerful and simulation techniques more refined. [1] At the same time, our understanding of the potential sources of error in a simulation has also grown. The computation of statistical errors in a MC run, and the way in which these errors depend on the length of the run as well as the correlation between successive configurations has been studied in detail, and there is now a well-developed formalism for determining errors in thermodynamic quantities calculated directly from the simulations. [2,3,4]
The development of histogram (reweighting) techniques [5,6] has allowed us to push the analysis of Monte Carlo data much farther than was previously thought possible. [7] Since the method is generally applicable to simulations of a wide variety of systems, the question of error determination in a histogram analysis is a problem of far ranging significance and interest. Results obtained by reweighting are subject to systematic errors due to the limited range of energies observed in a simulation, as well as an amplification of the normal statistical errors present due to the finite number of measurements made. In previous work, histograms from several different simulations (or equivalently different portions of a single, long simulation) were analyzed independently, and the variation of the results from the different analyses was used to estimate the error. While this is a practical and relatively efficient way to estimate errors (and in fact is the most commonly used method of error analysis in MC even when histogram reweighting is not used) it lacks the well-developed formalism which exists for standard MC simulations.
In this paper, we examine the question of the determination of the true statistical error of reweighted histogram data from a single run or histogram. In the following section we describe the theoretical formalism which can be used to determine the statistical error, and in Sections III and IV we present results for several simple cases which can be used to test this approach. We close with some general remarks in the final section.
