Student(s):
JaeAnn Dwulet
Dates of Involvement:
June 2009 - May 2010
Faculty Advisor(s):
Gunnar Martinsson
Graduate Mentor:
Patrick Young
Simulating Clusters of Charged Particles in Electric Potential Fields
Introduction
The research is aimed at developing techniques for simulating large clusters of particles trapped in a given potential well. A primary objective is to determine particle configurations that minimize the electric energy, but we are also interested in developing tools for performing dynamical simulations. Ideally, we wish to developed techniques that can handle very large number of particles in environments similar to those encountered in real world simulations of micro-biological processes.
Problem Description
Let 
denote a given potential function on or 
. Given a set of charges 
for 
, we consider two problems:
The "statics" or "energy minimization" problem: Find the positions 
such that the energy 
has a (global or local) minimum, where
The "dynamics" problem: Given a set of initial positions
, determine the trajectories of all charged particles as they move under the influence of electric forces from the other particles, and from the underlying potential. In other words, we seek to numerically solve the system of ordinary differential equations
Solution Techniques
Our initial work consisted of investigating and comparing different techniques for solving the statics problem. We first implemented Newton's method to find configurations where the gradient of W is zero. The situation where an energy minimizer is not a point but a whole manifold required some modified techniques as the Jacobian in this case becomes singular. With these modifications, the Newton method works extremely well when N is small, but the cost of evaluating the Jacobian matrix quickly becomes prohibitive. The secant method was also implemented, and was found to be capable of handling significantly larger problems.
We have also experimented with solving the statics problem by applying a steepest descent method from a random initial guess. In other words, we solve the differential equation
until we hit an equilibrium point. By generating a large number of random starting points, this method almost always finds what appears to be a global minimizer.
Preliminary Observations regarding Equilibrium Points:
• The equilibrium positions of the particles form a circle around the center of the potential well. The particle is at equilibrium at any point on the circle, but the other particles are at a distance far enough apart to keep the force minimized (Figure 1).
• Particles with a higher charge have equilibrium circles with larger radii because the force of repulsion with the other particles is greater (Figure 2).
• A geometric observation that was made was that in a symmetrical well the particles always form a symmetrical pattern in their equilibrium positions.
Future research directions
We are currently investigating how to efficiently solve the dynamics problem. Questions that must be addressed include finding a good time-stepping scheme for solving the ODE, and then to develop methods for handling near collisions.
The most costly component of our simulations is the evaluation of all pairwise electric force interactions. Since there are N*N pairs of N particles, naive techniques for evaluating all forces require O(N*N) operations as N grows. To combat this prohibitive growth, we plan to incorporate the Fast Multipole Method into our algorithms. This will keep the asymptotic complexity down to O(N).
Another objective of our research is to include dielectric interfaces into the model. Such models are common in biochemical simulations (where the interface could be, e.g., the boundary of a large organic molecule). We plan to do this using certain fast PDE solvers currently under development by Patrick Young.
We observe that the development of efficient techniques for solving the dynamics problem will also lead to efficient techniques for solving the statics problem via the steepest descent approach.
About JaeAnn Dwulet:
JaeAnn Dwulet is currently in her third year at CU-Boulder. She is an Applied Mathematics and Molecular, Cellular, and Developmental Biology double major. Her future plans include pursuit of a doctorate in Mathematical Biology.
References:
D. Griffiths, Introduction to Electrodynamics, 3rd Edition, Prentice Hall, New Jersey, 1999.
