Wolfram Computation Meets Knowledge

Wolfram Summer School


Jens von der Linden

Summer School

Class of 2010


Jens von der Linden is a graduate student at the Aeronautics & Astronautics Department of the University of Washington. At UW, he is studying plasmas, which are ionized gases consisting of a large number of neutral and charged particles. Although interactions between a few of these particles are relatively simple, in large numbers, their behavior becomes complex, with many spatial and temporal scales. Jens is developing cellular automata models of plasma relaxation. He enjoys Caribbean culture and food and regularly attends the West Indian Labor Day Parade in New York.

Project: Cellular Automaton Model of Taylor Relaxation in Plasmas

Magnetized plasmas naturally self-organize as magnetic fields relax to a minimum energy state while conserving helicity. Helicity is a measure of the extent to which the magnetic field lines wrap and coil around each other and is given by the volume integral of dot product of the magnetic vector potential with the magnetic field. Although the final relaxation state can be determined from the initial conditions, the dynamics of the relaxation process are poorly understood.

I propose to model the magnetic vector fields as directional links between lattice nodes. In 2D, the nodes can be linked to a maximum of four neighbors, and in 3D, they can be linked to a maximum of six neighbors. A physical constraint on the system is that the divergence of the magnetic field must be zero. This constrains the number of links each node can have to an even number. The number of node states is given by permutations of even numbers of linkages in all directions. To simulate the system, the lattice will be mapped to a cellular automaton with a neighborhood equal to the node neighborhood and number of colors equal to the possible states of linkages. The evolution rules of the system will be enumerated and analyzed with Mathematica code that will visualize the cellular automaton states as the connections in the lattice system. Directly calculating the overall helicity might prove difficult, as it is a continuous concept, but for simple configurations, it should be apparent from the visualization. To develop an intuition for setting up the system, I will start with a 2D model and then expand to 3D to fully capture helicity.

The goal of this project is to develop a better understanding of possible evolutions towards the relaxed helicity state. This will be achieved by searching the rule space for behavior resembling physical properties, such as the evolution to stationary states and helicity conservation.

Favorite Four-Color Totalistic Cellular Automaton

Rule 305171