The master of Monte Carlo

March 2010
Filed under: Lawrence Livermore

“It took one or two decades,” Alder says. “The spectacular results we got out of the computer methods eventually convinced most people.”

In the intervening years, Alder has developed a reputation for sniffing out and solving big problems in computational physics.

In the early 1970s, he and his LLNL collaborators used molecular dynamic methods to show that hydrodynamic properties are quantitatively applicable at less than nanoscale. The finding surprised people in the field because it was so novel.

“It wasn’t an experiment,” Alder notes, “and it wasn’t a theory. It was numerical work that first really overthrew a big principle in transport theory.”

Soon thereafter, he collaborated with physicist David Ceperley to solve what’s known as the “quantum many-body problem” using a Monte Carlo method to determine the properties of electron gases.   The problem had vexed physicists for decades because it contained a numerical instability, but the Monte Carlo method overcame that to find a solution for the homogenous electron gases.

The resulting publication, “Ground State of the Electron Gas by a Stochastic Method,” was recognized in 2002 as the third-most-cited Physical Review Letters paper ever. Other researchers, including density functional theorists who study chemical systems within the approximation of a uniform gas field, continue to apply the methods it described.

Not one to rest on his laurels – there are too many interesting questions still to be solved, he says – Alder is extending his work with Ceperley, now a professor at the University of Illinois.

The duo had to overcome the fermion problem when they set out to solve the quantum many-body problem using Monte Carlo methods.  Also known as the “sign problem,” the fermion problem is an inherent instability embedded in the solution to the Schrödinger equation, the essential quantum description of atoms’ behavior over time.

The issue occurs at the node, the place where the wave function changes from positive to negative. When the node is allowed to fluctuate, the amplitude of the wave grows at an exponential rate, and before long error in the system overcomes the true signal. To correct for this instability, Monte Carlo simulations don’t allow the nodes to move during calculations. The “fixed-node” simulations give reasonable approximations but with a fair amount of uncertainty in the answer.

Scientists can compensate for the uncertainty by building in experimental data on system energies, but they are still best guesses. The problem eluded even the famed theoretical physicist Richard Feynman, but, then again, he didn’t have the computational power available today. Alder dropped working on the problem in the 1970s, but recently took it up again.

“I’ve been waiting for 30 years for somebody else to do it,” he says. “But nobody has solved it, so I figured I’d better have another go at it. I want to make it work.

The project got its start over a weekly lunch with collaborator Randolph Hood, an LLNN condensed matter physicist who had worked on quantum Monte Carlo methods for years.

“Berni likes to look at the big picture,” Hood says. “He provides insights into the problem, and we try to implement them.”

Together with Norman Tubman, a graduate student in physics at Northwestern University, the group is making progress on the problem, but it’s slow going since Tubman is writing the code from scratch.

“If we can solve this problem, there would be a whole host of applications,” Hood says. “In any chemical reaction, if you know the energies, then you could calculate the transition energies and you could know the barriers for chemical reactions. In materials science, you would want to know the formation energy of a defect in the material. In nanotechnology it would be very advantageous to know accurate electronic energies.”

Alder says the group is working to model the first row of diatomic elements from lithium to fluorine to what he calls “chemical accuracy,” which would be two orders of magnitude better than previous methods.

“We hope to project out a solution before the instability takes over,” he says. “We’ve done lithium extremely well, and we are slowly moving up to the other elements.”

The group expects to submit a paper on this work this year.

“Just about every year someone publishes a potential solution to the sign problem,” Hood says. “But none of them ever work out.”

The group can’t say yet whether its solution will work.  But with Alder’s track record, they expect to push closer than anyone else.

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About the Author

Karyn Hede is news editor of the Nature Publishing Group journal Genetics in Medicine and a correspondent for the Journal of the National Cancer Institute. Her freelance writing has appeared in Science, New Scientist, Technology Review and elsewhere. She teaches scientific writing at the University of North Carolina, Chapel Hill, where she earned advanced degrees in journalism and biology.

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