Kay Kirkpatrick

Dilute Gases and Plasmas

To model dilute gases, where particles collide like billiards, there are two main "hard-sphere" models. First, there is the one-particle model, which looks microscopically like a billiard ball bouncing off large, round columns. The scaling limit consists of sending the diameter of the columns to zero, while increasing their density enough that the billiard doesn't just move freely. Second, and more interesting, is the full hard-sphere model, with all of the particles moving like billiards from an initial random configuration of positions and velocities. We can tag one particle and pass to the same kind of limit. In both models, it turns out that the particle's evolution is described macroscopically by the linear Boltzmann equation:
      ∂t f + v ⋅ ∇x f = Q(f, f) .
(Here Q is an integral operator encoding the collisions.) Some of my recent work provides a simpler, more elegant proof of this result. And these hard-sphere models are useful analogies for studying the soft-sphere models of plasmas.

The sun is a giant ball of plasma, producing energy through nuclear fusion (figure courtesy of NASA). In plasmas, the superheated, ionized particles experience grazing collisions, repelling each other strongly at short distances and weakly at long distances, as described by the Coulomb potential energy. This interaction potential has been notoriously difficult to handle in a scaling limit, so the strategy has been to study approximations instead, called soft-sphere models. The obstacles are still randomly placed, but now they look like hills instead of columns.

The one-particle soft-sphere model looks like a billiard ball grazing a bunch of gentle hills (seen from above on the left-hand side of the figure below). We can describe a variety of plasmas by introducing a parameter that measures the steepness and density of the obstacles (corresponding to the repulsive strength and density of a plasma's particles). One extreme of the parameter corresponds to the hard-sphere model, a low density of infinitely steep obstacles; the other extreme, a high density of slight obstacles. The scaling limit here, called the weak coupling limit, is to take the diameter of the obstacles to zero again, but now increase the density of obstacles faster. The Landau equation governs the macroscopic behavior in the weak coupling limit:
      ∂t f + v ⋅ ∇x f = ζ Δv f .
The particle travels in a velocity diffusion, a kind of smoothed-out Brownian motion, as depicted on the right-hand side of the next figure.

My dissertation extends and unifies the theory for the entire range of the parameter--especially in the middle range where obstacles overlap a lot and they influence the particle a lot. In particular, I prove that the diffusion coefficient, ζ, is independent of the parameter, meaning that the microscopic distinctions of the obstacles' steepness and density all disappear in the scaling limit, and the models have the same macro-behavior. Those are the one-particle soft-sphere models--we would really like to understand the full model, with all of the particles moving and grazing each other, but this has been intractable in the soft-sphere case. I hope, however, to adapt the new method for the full hard-sphere model to the multiple-particle soft-sphere model.

The real breakthrough (and one of my long-term goals) would be to handle the long-range Coulomb-interaction model. Previous results along these lines are incomplete and typically derived from hard-sphere methods; the Boltzmann equation is derived in the scaling limit, where we should expect the Landau equation. Soft-sphere methods may be a better approach to this problem, and computer modeling may help too.
(Figure: An experimental fusion reactor, empty and filled with a plasma. Courtesy of EFDA-JET.)
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