Efficient Thermodynamic Property Computation Using Molecular Simulation Over Thousands and Millions of Thermodynamic States

To design new molecules with specific thermodynamic properties under a range of external conditions
and compositions, researchers must be able to compute thermodynamic properties over large numbers, thousands
or even millions, of thermodynamic states. For complex molecules, simple equations of state are
insufficient to give reliable thermodynamic properties and full molecular simulations are the best way to
predict thermodynamic properties. In Monte Carlo simulations, thermodynamic properties are computed
for a given molecular system as averages over the configuration of the system for a single thermodynamic
state. If there are N possible parameters for our system, then if there are 10 possible choices for each parameter,
there are 10^N total ensembles that must be simulated. The problem is thus combinatorially large, and
it is impossible to directly perform 10^N separate simulations. By combining expanded ensemble techniques
and multistate reweighting methods, it is possible to treat external parameters, i.e. thermodynamic states, as
configurations of an expanded system and to simultaneously simulate over system configurations and possible
states [1, 2]. Expanded ensemble techniques such as the Wang-Landau algorithm involve Metropolis
Monte Carlo random walks in the allowed thermodynamic states, or state space [1]. Multistate reweighting
methods such as the Multistate Bennet acceptance ratio (MBAR) involve gathering statistical information
determined by one set of molecular parameters using data from another set of molecular parameters [2]. As
we show, using these algorithms jointly makes possible the computation of thermodynamic averages over
nearby states in simulations where the number of samples collected is less than the number of possible states.
The goal of this research is to understand how to efficiently sample large numbers of states simultaneously
in order to improve the in silico design of nanoscopic materials.

In this study, we examine joint simulations over atomic spin configurations and external magnetic fields
in the 2D Ising model. This model was chosen because it is well studied, holds physical significance in
materials design, and allows for a large number of possible thermodynamic states [3]. Thermodynamic
states in this case are defined by the external magnetic field, which can be different at every position on the
Ising lattice. We compare the simulation time it takes to reach a target statistical uncertainty in the average
spins and free energy for 10, 1000, 10000 and 10⁶ possible configurations of the external magnetic field
by both fixed-state sampling and by the use of two expanded ensemble techniques. The efficiency of the
Wang-Landau algorithm and a weighted version of the Wang-Landau algorithm are compared with explicit
reweighting methods. Temperature dependent properties are also studied.

References
[1] John D Chodera and Michael R Shirts. Replica exchange and expanded ensemble simulations as
Gibbs sampling: simple improvements for enhanced mixing. The Journal of chemical physics,
135(19):194110, November 2011.
[2] Michael R Shirts and John D Chodera. Statistically optimal analysis of samples from multiple equilibrium
states. The Journal of Chemical Physics, 129(12):124105, September 2008.
[3] Lars Onsager. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Physical
Review, 65(3-4):117–149, February 1944.