Seminar #6 - Enhanced Monte Carlo

2024 Winter Molecular Simulation Seminar
Room 302, the 2nd experimental building at POSTECH
Presenter: Sangmin Lee

The MC method is based on the idea that a determinate mathematical problem can be treated by finding a probabilistic analogue which is then solved by a stochastic sampling experiment. Along with the MD simulation, MC method is widely applied in various molecular systems to sample the equilibrium ensembles such as isobaric ensemble, canonical ensemble, grand-canonical ensemble, osmotic ensemble, and even Gibbs ensemble. The main difference between MC and MD is that although both methods are aimed to generate a trajectory in phase space which samples from a chosen statistical ensemble, the evolution to sample the states of the system is made by integrating the equation of motion in MD simulation while in MC simulation, that is made by accepting or rejecting the chosen random walk based on Metropolis algorithms. Although MC simulation is a powerful tool for studying the equilibrium properties of matter, it was progressively realized that conventional MC really requires more substantial sampling over high-energy configruations to accurately compute the ensemble averages. For instance, in complex condensed-phase systems, it is difficult to design MC moves with high acceptance probabilities that also rapidly sample uncorrelated configurations. These obserations clearly indicated that improved MC moves and algorithms are required to encourage the system to explore regions of phase space not frequently sampled by the Metropolis algorithms.

In this talk, I'll discuss the theoretical framework to extend the repertoire of useful MC moves to guide the system in its search for favorable configurations by avoiding barriers in phase space. One way is to change the underlying stochastic matrix of Metropolis algorithms to make the MC 'smarter' [1] at choosing its trial moves in condensed-phase system. The other way is to introduce the idea of expanded ensembles[2], which is a set of sub-ensembles linked by a coupling parameter. By changing the value of coupling parameter in an expanded ensemble, one can perform a random walk between the corresponding sub-ensembles and calculate thermodynamic properties with high acceptance probabilities. I'll discuss the development of MC algorithms that are specifically aimed to sample the expanded ensemble, the Continuous Fractional Component MC (CFCMC) [3,4] and Nonequilibrium Candidate MC (NECMC)[5,6]. Introducing some of the character of MD into MC simulations (hybrid MCMD [7,8]) may also be advantageous, particularly if collective motions are important in avoiding barriers in phase space.

Reference

[1] Senderowitz, H., Guarnieri, F., & Still, W. C. A smart Monte Carlo technique for free energy simulations of multiconformational molecules. Direct calculations of the conformational populations of organic molecules. J. Am. Chem. Soc., 1995, 117(31), 8211-8219.
[2] Lyubartsev, A. P., Martsinovski, A. A., Shevkunov, S. V., & Vorontsov‐Velyaminov, P. N. New approach to Monte Carlo calculation of the free energy: Method of expanded ensembles. J. Chem. Phys., 1992, 96(3), 1776-1783.
[3] Shi, W., & Maginn, E. J. Continuous fractional component Monte Carlo: An adaptive biasing method for open system atomistic simulations. J. Chem. Theory. Comput., 2007,3(4), 1451-1463.
[4] Rahbari, A., Hens, R., Ramdin, M., Moultos, O. A., Dubbeldam, D., & Vlugt, T. J. H. Recent advances in the continuous fractional component Monte Carlo methodology. Mol. Sim., 2021, 47(10-11), 804-823.
[5] Nilmeier, J. P., Crooks, G. E., Minh, D. D., & Chodera, J. D. Nonequilibrium candidate Monte Carlo is an efficient tool for equilibrium simulation. PNAS, 2011, 108(45), E1009-E1018.
[6] Ross, G. A., Rustenburg, A. S., Grinaway, P. B., Fass, J., & Chodera, J. D. Biomolecular simulations under realistic macroscopic salt conditions. J. Phys. Chem. B, 2018, 122(21), 5466-5486.
[7] Barhaghi, M. S., Crawford, B., Schwing, G., Hardy, D. J., Stone, J. E., Schwiebert, L., ... & Tajkhorshid, E. Py-MCMD: Python software for performing hybrid Monte Carlo/molecular dynamics simulations with GOMC and NAMD. J. Chem. Theory. Comput., 2022, 18(8), 4983-4994.
[8] Neyts, E. C., & Bogaerts, A. Combining molecular dynamics with Monte Carlo simulations: implementations and applications. Theoretical Chemistry in Belgium: A Topical Collection from Theoretical Chemistry Accounts, 2014, 277-288.