Optimal estimation of free energies and stationary densities from multiple biased simulations

TL;DR

The paper introduces TMU, a new method for estimating free energies and stationary densities from biased simulations that relaxes previous assumptions, enabling analysis of high-dimensional systems with unknown energy landscapes.

Contribution

The paper presents TMU, a novel estimation technique that requires only local equilibrium and can handle high-dimensional state spaces without predefined order parameters.

Findings

TMU converges asymptotically and is normally distributed.

TMU performs well on numerical examples, demonstrating its effectiveness.

The method simplifies free energy estimation in complex systems.

Abstract

When studying high-dimensional dynamical systems such as macromolecules, quantum systems and polymers, a prime concern is the identification of the most probable states and their stationary probabilities or free energies. Often, these systems have metastable regions or phases, prohibiting to estimate the stationary probabilities by direct simulation. Efficient sampling methods such as umbrella sampling, metadynamics and conformational flooding have developed that perform a number of simulations where the system's potential is biased such as to accelerate the rare barrier crossing events. A joint free energy profile or stationary density can then be obtained from these biased simulations with weighted histogram analysis method (WHAM). This approach (a) requires a few essential order parameters to be defined in which the histogram is set up, and (b) assumes that each simulation is in global equilibrium. Both assumptions make the investigation of high-dimensional systems with previously unknown energy landscape difficult. Here, we introduce the transition matrix unweighting (TMU) method, a simple and efficient estimation method which dismisses both assumptions. The configuration space is discretized into sets, but these sets are not only restricted to the selected slow coordinate but can be clusters that form a partition of high-dimensional state space. The assumption of global equilibrium is replaced by requiring only local equilibrium within the discrete sets, and the stationary density or free energy is extracted from the transitions between clusters. We prove the asymptotic convergence and normality of TMU, give an efficient approximate version of it and demonstrate its usefulness on numerical examples.