Calculation of free energy through successive umbrella sampling

TLDR

The study introduces a consecutive‑window umbrella sampling scheme that eliminates the need for a pre‑defined weight function and demonstrates that its error is controlled and independent of window size. The method partitions the state space into overlapping windows, samples them sequentially, and uses the results to estimate the weight function, enabling detection of sampling issues and equilibration, as illustrated by a Lennard‑Jones liquid‑vapor simulation. The implementation achieves error control independent of window size and attains efficiency comparable to multicanonical simulations without requiring a pre‑determined weight function.

Abstract

We consider an implementation of umbrella sampling in which the pertinent range of states is subdivided into small windows that are sampled consecutively and linked together. This allows us to simulate without a weight function or to extrapolate the results to the neighboring window in order to estimate a weight function. Additionally, we present a detailed error analysis in which we demonstrate that the error in umbrella sampling is controlled and, in the absence of sampling difficulties, independent of the window sizes. In this case, the efficiency of our implementation is comparable to a multicanonical simulation with a very good weight function, which in our scheme does not need to be known ahead of time. The analysis also allows us to detect sampling difficulties such as correlations between adjacent windows and provides a test of equilibration. We exemplify the scheme by simulating the liquid-vapor coexistence in a Lennard-Jones system.