AI Helps Researcher Find Exact Solutions to Longstanding Fusion Physics Problem

In 1967, physicist Harold Grad questioned whether smooth toroidal equilibria with varying pressure generally exist without additional symmetry. The conjecture and related formulations that emerged from this have been a subject of mathematical research ever since.

Landreman's paper provides a counterexample to one formulation of the conjecture, but the paper also notes that recent work has already disproved the conjecture by establishing the existence of another family of asymmetric solutions. Therefore, the new contribution is more specific than being the first to solve the problem: he provides two additional families of solutions whose magnetic fields and pressure can be written explicitly using elementary mathematical functions.

Such configurations are significant for stellarators, which rely on complex magnetic geometry to confine high-temperature charged gas plasma. The paper shows that strongly asymmetric toroidal magnetic field configurations can maintain fully nested magnetic surfaces while balancing plasma pressure.

The two families of solutions differ in how magnetic field lines wind around nested magnetic surfaces. In the first family of solutions, each magnetic field line goes around the toroidal cross-section twice for every one time it goes around the central opening, and the rotational transform remains the same on all magnetic surfaces; this family is constructed by stretching an initially rotationally symmetric magnetic field.

In the second family of solutions, the rotational transform changes from one magnetic surface to another, meaning magnetic shear is present, with some magnetic field lines closing and others not closing. This extends the mathematical examples beyond configurations with a single fixed rotational transform.

The paper derives the resulting magnetic surfaces and shows that the magnetic field, current density, and pressure remain smooth within the chosen toroidal region, with the pressure gradient vanishing only on the magnetic axis. Both families of solutions contain adjustable parameters and are groups of solutions rather than isolated examples.

Landreman reports that these solutions were discovered using GPT-6 Astra Pro, which also helped draft portions of the paper. He writes that he manually verified every equation.

A distinction needs to be made: the existence and properties of the two families of exact solutions are mathematical results presented in the paper; the specific degree of GPT-6 Astra Pro's contribution to the discovery process comes only from Landreman's own account, with no third-party verification. The study is posted on the preprint server arXiv and has not yet undergone peer review.

According to Landreman, these exact results can serve as benchmarks for numerical equilibrium codes. Researchers can use software to compute configurations and compare against known mathematical answers to identify errors or assess accuracy.

The paper also provides quantities for comparison calculations, including average field strength, pressure, and the ratio of plasma pressure to magnetic pressure. The source mentions that Landreman points to supplementary scripts for comparing numerical equilibrium calculations with exact solutions, but the source does not provide specific data from these scripts and numerical comparisons.

The same mathematics also describes steady-state flow of certain ideal incompressible fluids, extending the significance of this finding beyond fusion physics.

This theoretical study does not report fusion experiments, increased energy output, or a buildable reactor design, nor does it prove that the physical plasma in these configurations remains stable when perturbed, is sufficiently well insulated, or can actually be built with engineering magnets.

Exact equilibrium is only one condition for a useful fusion configuration. The existence of these particular solutions does not mean that every desired asymmetric configuration can support perfectly nested magnetic surfaces; the formulas address a specific existence problem, and broader design questions remain open.

Whether these mathematical solutions can be generalized to more general asymmetric configurations remains unclear, and their applicability to the engineering design of actual fusion devices has not been established.

Landreman points to several directions for follow-up research, including finding solutions with different numbers of repeating magnetic segments and other winding modes of magnetic field lines, as well as more solutions that do not preserve the symmetry of the examples in this paper.

The source also mentions that other questions involve whether similar explicit solutions exist under different magnetic confinement conditions, including current-free magnetic fields within the region studied. Whether these directions can produce equally explicitly writable solutions, the source does not judge.