Journal
MULTISCALE MODELING & SIMULATION
Volume 12, Issue 1, Pages 25-54Publisher
SIAM PUBLICATIONS
DOI: 10.1137/120895883
Keywords
Markov chain; maximum likelihood estimation; error analysis; free energy; stationary density; simulation
Funding
- DFG [NO 825/2-1, WU 744/1-1]
- DFG research center Matheon
- ERC [307494 pcCell]
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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 the estimation of the stationary probabilities by direct simulation. Efficient sampling methods such as umbrella sampling, metadynamics, and conformational flooding have been developed that perform a number of simulations where the system's potential is biased so 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 the weighted histogram analysis method. 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 based unbiasing method (TMU), 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 a preselected 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 in numerical examples.
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