A well-known pragmatist among theoretical physicists, Berni Alder’s first thought after hearing last fall that he was to receive the National Medal of Science, the nation’s highest honor for scientific achievement, was that maybe this would attract more people to work on projects with him.
Although officially retired from Lawrence Livermore National Laboratory (LLNL), Alder has a laundry list of unsolved issues in molecular dynamics, the computational method he invented and which has garnered him international accolades. Molecular dynamics and Alder’s Monte Carlo methods are nearly ubiquitous, used to solve problems in physics, molecular biology and other diverse areas.
After such a distinguished career, Alder certainly could have retired with his place secure in the history of computational physics. But he still feels the same need to prove his methods as he did in the early 1950s, when the idea of using computational methods to model physical systems was controversial.
“Among theoretical physicists in the early days there was great bias against computational methods, because they are cumbersome,” Alder says. “You want a nice analytic solution. It took decades to get to computational methods accepted intellectually by the hard-nosed theoretical physicists.”
Alder began as one of those theoretical physicists, but the complexity of the problems he was trying to solve drove him to computational approaches. As a graduate student, Alder worked with theoretical chemist John Kirkwood at the California Institute of Technology on the fundamental question of whether “hard spheres,” an idealized representation of the atoms, could form a solid without attractive forces between the spheres. If you could pack molecules more efficiently in a solid than a fluid, would their atom-spheres condense into a solid? And what would that phase transition look like?
Kirkwood had been working on a theoretical answer to the question, but it required the solution of a nonlinear integro-partial differential equation – notoriously difficult equations for finding an analytical solution and determining whether it’s unique.
“We didn’t know whether a solution existed,” Alder says.
The problem required a large number of calculations repeated over and over. The year was 1948, and the U.S. nuclear arms race with the Soviets was reaching fever pitch. The Los Alamos and Lawrence Livermore weapons labs had developed computational methods to handle complex nuclear reactions and some of those chemists and physicists now were moving into academic roles. Alder asked one such physicist, Stan Frankel, then head of a new digital computing group at Caltech, to help him with his hard sphere problem.
The two pondered how to solve the integral equation before seeking a better way.
Frankel had encountered computational modeling techniques such as the Monte Carlo method at Los Alamos, where he worked on the Manhattan Project. Before long, he and Alder were applying the method to solve Alder’s problem. It soon became apparent that the large computers needed to solve these problems resided at the national labs, and Alder took a position at LLNL, teaming with physicist Thomas E. Wainwright to complete the hard sphere problem.
The hard-sphere transition still cannot be solved using theoretical methods. If Alder and colleagues had continued working the problem using the old methods, they would almost certainly have failed. Instead, they ushered in a technique scientists and engineers still use today to understand the atomic and molecular interactions in many chemical and biological systems.
The 1956 paper that emerged from these early forays, “Phase Transition for a Hard Sphere System,” is often marked as the birth of molecular dynamics as a field.
Molecular dynamics takes advantage of what computers are good at: solving a given calculation in repeating loops. Given the precise coordinates of every atom in a defined system, the effect of hard sphere collisions can be calculated over time. The genius of Alder’s method was circumventing uncertainty, the bugbear of partial differential equations, and finding a numerical solution with no approximations.
“Computational methods are like numerical experiments,” he says. They enable scientists to describe a system mathematically, then perturb it in some way and allow the numerical experiment the freedom to react to those perturbations.
“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.