In May 2021, Princeton University has ended **Very detailed research work**with the aim of determining (or at least trying to do so) the ideal geometry for a nuclear reactor, the “box” in which it is **The plasma reaches temperatures high enough to trigger nuclear fusion**the same physical process that occurs every moment on the stars.

The paper is titled ** On quasi-symmetric plasma equilibrium supported by a small force.** Published in February 2021 in The Journal of Plasma Physics.

The Princeton work analyzed the pros and cons of the two reference models par excellence, the tokamak on the one hand and the more modern model on the other.

The former has been the dominant model for several decades and has taken advantage of the simplicity of its structure to allow scientists to focus on the search for confinement; but, **Its biggest strength turns out to be its biggest drawback, because its shape creates significant engineering challenges for stabilizing the plasma**. Also, current models** It consumes a lot of energy** In order to be energy efficient.

**The Stellarator was created, as it is easy to guess, to solve the shortcomings of the tokamak, replacing axial symmetry with a twisted exoskeleton**. The resulting fluctuation resolved many of the predecessor’s shortcomings, while also opening up the research to new insight: *Would it be possible to create “custom” magnetic fields within the stellar geometry, to reach a stable confinement of plasma?*

These stable configurations are called “equilibrium” and their feasibility is closely related to the study of the reactor itself. Therefore, research has begun to follow this path with the aim of research **Alternative shapes that use new engineering properties to overcome the tokamak’s major energy efficiency problems, while remaining simple enough to realistically study and build.**

Mathematicians Peter Constantine and Daniel Ginsberg of the Princeton Department of Mathematics, working with Theodore Drivas of Stony Brook University, reconsiders how flexible the geometry of stars is.

Their analytical perspective allows the application of tools from differential geometry and differential equations *[ndr, sottocampi della ricerca matematica pura]* Demonstrate the existence of different methods for finding magnetic equilibrium. It may seem like a trivial finding, but this study actually proves it *“The physics of nuclear fusion has a deep mathematical framework, which can be exploited to find a whole ‘zoo’ of possible new star designs.”*.

### Mathematics, Engineering and Physics

To understand the complexity of this tipping point, let’s start from the fact that the physical equations governing magnetically charged plasmas are a system of nonlinear partial differential equations (PDEs). PDEs are a mathematical language used to study the dynamics of sophisticated physical systems, which include a major branch of mathematics and physics. Nonlinearity means that the terms of an equation interact with each other and themselves, and this often leads to unpredictable behaviour.

*“By defining our problem in the language of physics and PDEs, we model the behavior of our plasma as a gaseous fluid moved by large magnetic fields. The balance we seek is the self-sustaining magnetic fields present within the volume of the plasma.”*In a note, the authors explain the use of magneto-hydraulic equations (MHS) to find magnetic equilibrium.

The list of conditions goes on, and any hope of optimizing stellar geometry to allow for neighboring magnetic field solutions depends on a careful balance of physics, geometry, and mathematical analysis.

However, the authors’ analysis continues by attacking a simpler problem. They don’t take completely general simplification tools and ask for solutions to the above MHS equations. Instead, they took inspiration from the plasma physics literature to search for a simpler subclass of solutions, called nearly symmetric equilibrium solutions.

For a better understanding, let’s remember that a tokamak is a very symmetric reactor design. In particular, it has an orientation (angular orientation) **along which the geometry is the same, and thus solutions of the previous MHS equations respecting this symmetry can be sought**. In practice, this means that instead of solving the entire 3D problem for magnetic field as a function of x, y, and z coordinates, **Symmetry is used to suppress one direction and solve the problem of dimensionality reduction.**

The main result of the article which is formulated as a mathematical theory and roughly states that, under certain mathematical assumptions, it is possible to solve MHS equations in general geometries and find magnetic equilibrium. More precisely, the authors show that, starting with a tokamak, its geometry can be distorted (for a familiar image, consider taking donut dough and squeezing, pulling, and rolling it).** Finding a magnetic balance even in the new shape of the reactor.**

Professor Drivas explained this in his conclusions *“In principle, our theory provides a constructive method by which quasi-equilibrium plasma configurations with good confinement properties can be constructed.”*.

However, he cautions, building stars capable of producing these configurations remains a difficult mathematical and engineering challenge.

### The departure of the stick and the future of fusion

Challenge met: recently, in fact, a second group of researchers from Princeton, based on these two works, identified *“abbreviation”* Mathematics to find the best way to harness fusion energy.

The method allows scientists Dr**I predict the star’s ability to retain heat by measuring the ability of the reactor’s magnetic field to retain fast-moving atomic nuclei in the plasma**, which increases the total heat and aids fusion reactions. This will allow to find the mathematically optimal shape of the star.

*“We cannot simulate the motions of all individual particles in all possible magnetic fields: that would require nearly unlimited computing power.”*said Alexandra Levines, a graduate student in plasma physics at the Princeton Plasma Physics Laboratory (PPPL) of the US Department of Energy (DOE). *Instead, we can use an acronym.*added the researcher.

*This research shows that Finding the best magnetic field shape for heat trapping is within our grasp, or perform calculations in general, or how far fast particles move away from the curved surfaces of the magnetic field in the center of the plasma. This behavior, described by a number, is known as C bandwhich we found to correspond consistently with plasma confinement”*.

The central point can be summarized as follows: **The more fast-moving particles left in the plasma medium, the hotter the fuel and the more efficient the star apparatus**.

Once this hypothesis has been established, the next step is to understand that rather than calculating the trajectory of each particle individually (a mathematically daunting task), one can only calculate how these particles are repelled by magnetic fields.

LeVilness’ hunch **Not only has it proven itself, it has already been applied**explained Elizabeth Ball, assistant professor of applied physics at Columbia University.

*“Using techniques like the one studied by LeViness, we’ve been able to find magnetic configurations of stars that have heat very much like tokamak. It’s challenging, but possible.”*.

### Read more

To learn more about the topic and get an updated overview of the latest nuclear fusion techniques, here is the list of articles written so far:

–Nuclear fusion: in eight years he can power up the grid;

– magnetic fields bring us one step closer to nuclear fusion;

– Nuclear fusion in the ancient continent, where are we?

Nuclear Fusion: The UK wants to build a prototype;

– Nuclear fusion: Japanese research will help ITER, but only if it lasts;

– A star is born: it is called Proxima Fusion and it will bring nuclear fusion to Europe within the next seven years;

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