Toolbox – Monte Carlo method

    Click to enlarge. Credit: Amy Orsborn

    Our physical world feels deterministic. A billiards player depends on this when she carefully lines up complex shots across several cushions. Make the billiard balls ten thousand times smaller and play underwater, however, and she would have a much tougher time predicting the shot’s outcome (let alone finding a very tiny pool cue). Indeed, the behavior of molecular-scale systems, especially molecules in liquid solvents, is often best represented as a series of random events called a random process. The modern study of molecular random processes began in the late 19th century, when Austrian scientist Ludwig Boltzmann discovered his famous equation that connects the laws of physics to the laws of probability.

    Studying random processes with equations can be fiendishly tricky, however, and this approach has limits like any other research tool. Would it be possible to take the same mathematical model and simulate its random behavior on a computer? Such was the thinking at Los Alamos National Laboratory among the scientists that were in charge of nuclear weapons development during and after World War II (better known as the Manhattan Project). In the method they devised, random numbers generated on a computer are used to choose new positions for a simulated set of molecules. These new positions are either accepted or rejected based on Boltzmann’s equation. As this process repeats, the model traces out molecular configurations according to the laws of thermal physics. Mathematical physicist John von Neumann code-named this method “Monte Carlo,” in reference to the town along the French Riviera where games of chance are played on this casino gambling site and in real life casinos. Intriguingly, when the method was first published in 1953, the paper’s final author was Edward Teller, “the father of the hydrogen bomb,” making the Monte Carlo technique, somewhat awkwardly, the bomb’s more benign older sibling.

    The Monte Carlo method has been an invaluable tool throughout the physical sciences ever since. Where non-computational methods might require approximations, Monte Carlo techniques can provide a way to attain exact numerical calculations for the various properties of a given theoretical model. In 1970 it was shown that the method could be generalized to probabilistic models far beyond those used in physics. Since then the application of this method has spread throughout the fields of economics, genomics, sociology, and more. Indeed, its power lies in its flexibility. Researchers can use any computational “move” (i.e., a way to generate the next step in the random process) that they can dream up, as long as the move obeys Boltzmann’s equation or similar criteria. New types of Monte Carlo moves can help computational scientists study research problems with ease that would be impossible, or at least prohibitively slow, using straightforward Monte Carlo moves.

    Berend Smit and coworkers at UC Berkeley and LBL develop and apply Monte Carlo techniques to study porous materials like zeolites and metal-organic frameworks (MOFs). These materials can capture gases like carbon dioxide (CO2) before they are released into the atmosphere and are thus a promising clean energy technology. However, they are very challenging to study experimentally. Monte Carlo techniques have therefore provided an especially valuable way to understand their characteristics.

    How would you design a simulation to study how CO2 molecules are absorbed by a MOF? Perhaps you would run a Monte Carlo simulation where the gas molecules start outside a block of material and gradually wander inside—but these simulations take far too long to run. To circumvent this problem, Smit and coworkers had to develop new, physically-justified moves that would allow gas molecules to be inserted or deleted throughout the structure based on their interactions with the surrounding MOF environment, all in a computationally efficient way. These so-called “grand canonical” moves give the same answer as the simple diffusion-based simulations would, and they can get there much faster.

    James Bond, the fictional master of myriad specialized gadgets, was also inspired by the seaside casino that von Neumann had in mind. One could argue however, that Monte Carlo techniques empower you to be more cunning than even a secret agent. You are often the maker of the tools as well as the master, deftly addressing each perhaps-unexpected, apparently-insurmountable challenge in your path. Not bad for the sibling of a nuclear weapon.

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