Bringing to Earth the virtually unlimited fusion energy that powers the sun and stars requires modeling the hot, charged plasma gas that fuels fusion reactions. Now scientists at the U.S. Department of Energy’s (DOE) Princeton Plasma Physics Laboratory (PPPL) have discovered a new theoretical method for designing computer algorithms to model the fuel and speed the development of a safe and clean energy source for humanity.

“Our research rigorously demonstrates how to preserve fundamental mathematical properties of plasmas in simulation algorithms,” said Alexander Glasser, a graduate student in the Princeton Program in Plasma Physics at PPPL and lead author of a paper in the Journal of Plasma Physics that lays out the findings. “We provide a blueprint for the construction of algorithms that more faithfully describe the real world.”

Fusion combines light elements in the form of plasma — the state of matter composed of free electrons and atomic nuclei that makes up 99 percent of the visible universe — to produce vast amounts of energy. Scientists around the world are seeking to duplicate this process to generate electric power.

The new method of algorithmic design focuses on preserving the underlying mathematical structures of plasma systems. One such structure preserved by Glasser and physicist Hong Qin, his coauthor and advisor, is known as “gauge structure,” which describes the symmetries and conservation laws of a physical system.

Their research derives from a theorem developed in 1915 by German mathematician Emmy Noether. According to her theorem, the “gauge symmetry” of electromagnetic systems gives rise to their conservation of electric charge.

In its original form, Noether’s theorem applies to physical systems described by differential equations. But since differential equations of plasma systems can be challenging to solve, physicists use computer algorithms that only roughly approximate the equations when modeling plasma dynamics. Using conventional methods, it is difficult to design algorithms that conserve charge as the differential equations do.

In this new finding, Glasser and Qin show that properly constructed algorithms can in fact preserve key mathematical features of the differential equations, including electric charge. The new computer algorithms conserve this property with the highest precision possible on computers.

The algorithms describe plasmas using Lagrangian and Hamiltonian mechanics — standard formulations of classical mechanics developed in the 18^{th} and 19^{th} centuries. The formulations led Glasser and Qin to discover a general class of algorithms, technically called “gauge-compatible splitting methods,” which preserve gauge structure and conserve charge in Hamiltonian systems.

The new algorithms also preserve what is called the geometric “symplectic structure” of plasma systems. As a result, these algorithms also conserve what is known as the density of phase space, or roughly the number of states that a plasma system can be in.

The proposed algorithm now requires implementation in a simulation code that models a fusion facility. “The design proposed in this paper is guided by the philosophy that space-time is discrete and not continuous, and all the laws of physics can be established on discrete space-time,” Qin said. “It turns out that the techniques enabled by this philosophy can lead us to more accurate and reliable simulation algorithms for fusion energy devices.”

Support for this work comes from the DOE Office of Science (FES) and the Princeton University Charlotte Elizabeth Procter Fellowship.

PPPL, on Princeton University's Forrestal Campus in Plainsboro, N.J., is devoted to creating new knowledge about the physics of plasmas — ultra-hot, charged gases — and to developing practical solutions for the creation of fusion energy. The Laboratory is managed by the University for the U.S. Department of Energy’s Office of Science, which is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of the most pressing challenges of our time. For more information, visit energy.gov/science.

*PPPL is mastering the art of using plasma — the fourth state of matter — to solve some of the world's toughest science and technology challenges. Nestled on Princeton University’s Forrestal Campus in Plainsboro, New Jersey, our research ignites innovation in a range of applications including fusion energy, nanoscale fabrication, quantum materials and devices, and sustainability science. The University manages the Laboratory for the U.S. Department of Energy’s Office of Science, which is the nation’s single largest supporter of basic research in the physical sciences. Feel the heat at https://energy.gov/science and https://www.pppl.gov. *