Linking doughnuts, soda straws and energy.
Linking doughnuts, soda straws and energy
Knowing how magnetic field lines in a neutron star neutron star, extremely small, extremely dense star, about double the sun's mass but only a few kilometers in radius, in the final stage of stellar evolution. Astronomers Baade and Zwicky predicted the existence of neutron stars in 1933. are tangled or linked gives scientists a good estimate of the minimum energy such a system can have. That surprising result is one consequence of a recently proved mathematical theorem linking the topology, or geometric form, of lines representing the flow of an incompressible in·com·press·i·ble
Impossible to compress; resisting compression: mounds of incompressible garbage.
in fluid (whether an ideal liquid or a magnetic field) and the energy embodied in that fluid flow.
"What's of interest is the relationship between the topology of the lines and the energy in the system," says mathematician Michael H. Freedman of the University of California, San Diego UCSD is consistently ranked among the top ten public universities for undergraduate education in the United States by U.S. News & World Report. It is a Public Ivy.  For graduate studies, most of UCSD's Ph.D. .
One way to picture the situation is to think of a solid torus In mathematics, a solid torus is a topological space homeomorphic to , i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. , or doughnut sh ape, made up of stretched rubber bands, each running once around the doughnut's hole. The idealized i·de·al·ize
v. i·de·al·ized, i·de·al·iz·ing, i·de·al·iz·es
1. To regard as ideal.
2. To make or envision as ideal.
1. rubber bands want to shrink as much as possible, but because the material is also incompressible, they can't make the doughnut shrink to nothing. However, the rubber bands can release their energy by forcing the doughnut's shape into that of a long, thin soda straw. Like the doughnut, the soda straw has a central hole, but because of the straw's much narrower configuration, each rubber band has a significantly shorter distance to stretch.
When one doughnut winds through the hole of a second doughnut, the situation changes dramatically. The soda-straw solution is no longer possible. If one doughnut were to spring into its soda-straw configuration, the change would stretch the other doughnut in a way that increases the energy of its rubber bands. In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently , the linking of the two doughnuts prevents the dissipation of energy Same as
See also: Dissipation . Similar argunents hold for more complicated links and knots.
In the case of a neutron star, the material within the star moves around so as to let magnetic field lines straighten out and separate as much as possible. But any linking and knotting of these lines gets in the way, limiting how much the system can "relax" and setting a lower bound on its energy. Similarly, a solid torus of spinning fluid may give up energy by elongating like a soda straw, but such a shift is prevented when several tori link.
Freedman, working with graduate student Zheng-Xu He, builds on a theorem proved by Soviet mathematical physicist V.I. Arnold, who showed that energy bounds exist for the special case when a quantity called the "linking number" can be computed for a given tangle of lines. Using ideas from knot theory knot theory
Mathematical theory of closed curves in three-dimensional space. The number of times and the manner in which a curve crosses itself distinguish different knots. , Freedman and He extended Arnold's theorem to cover a much broader range of knots and links.
"Geometric linking, independent of linking number, can be used to estimate a lower bound on the energy of an incompressible flow Incompressible flow
Fluid motion with negligible changes in density. No fluid is truly incompressible, since even liquids can have their density increased through application of sufficient pressure. ," Freedman says. "This suggests wider applicability of the topological lower-bound principle than was visible from Arnold's work."
The new results provide an intriguing hint of how to keep a hot, ionized i·on·ize
tr. & intr.v. i·on·ized, i·on·iz·ing, i·on·iz·es
To convert or be converted totally or partially into ions.
i gas, or plasma, from leaking out of a magnetic "bottle" -- a problem faced in fusion-energy research. In this case, creating local "tangles" in a confining magnetic field many keep energy from dissipating below a certain level.
"Any real-world system is much more complicated than what we addressed," Freedman says. "But that's often the nature of mathematical physics. You don't try to build a model as complicated as the world. You try to build something you can analyze, and then you take it from there."