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On the Diusion of Physical Processes in Radially

Distributed Systems

Alfred Hubler and Sam Foreman

Abstract

We begin by studying the diusion process on a uniform angular distributionof points, whose radial density varies as 1

r2 . We then go on to investigate

how the diusion rate (namely, the movement of the center of heat) varieswith dierent distribution functions. In order to make our model realistic, weproceed to transfer the substance to all neighbors (points) within a predefined,fixed radius. We refer to those points for which our substance (heat, infection,sound, etc.), has already diused to as being active. This process runs for afixed number of iterations, where each iteration randomly selects a point thatis active, and again diuses to all neighbors within the given radius. In order toget a better understanding of the chaotic dynamics exhibited by our system, wealso compute and plot the centroid for each iteration. The goal of this researchis to perform a dimensional reduction from a system with multiple variables

into a lower-dimensional description with a smaller number of free parameters.

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Introduction

Inverse Square Distribution

The program begins by defining a fixed radius (presently, 0.20 units), and thengoes on to construct arrays and r which are distributed uniformly, and as theinverse square of the radius, respectively. Next, we select a point (x, y) ! (r,)at random which we use as our initial source (~r0). The system then begins topropagate radially by distributing the heat to all neighboring points within the givenradius (r = 0.20) (i.e. a point P is defined as active if k~r(P) ~r0k r

) Thisis represented graphically by coloring the active points red. Next, we computethe centroid for a given iteration by taking the weighted average of the activepoints distance from the previous iterations centroid, and marking it with a bluestar. Finally, we create a plot of the system for each time step, to show how oursystem evolves in time. This process is then repeated, for N = 50 iterations. Forthe last iteration, we compute the final center of mass, marked by a green star.Finally, we run a statistical analysis on both the angular and radial distribution ofpoints, and plot these accordingly. We complete this analysis by creating a best-fit polynomial for our actual points, and compute the absolute error between thisbest-fit polynomial, and the expected results, as shown.

Figure 1: Angular Distribution Figure 2: Radial Distribution

Where the angular distribution plot represents i = tan1(xi

yi) vs. (xi, yi), and the

radial distribution histogram represents the number of points vs. radial distance.

Note that the physical quantity of interest is the rate of diusion, and in particular,the movement of the center of active points, ~rcm relative to the center of the wholesystem, defined as the sum of each points distance from the origin, over the totalnumber of points, and denoted by ~Rcm.

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We plot this as a function of iteration number, k, as shown.

Figure 3: k~rcm ~Rcmk vs. k

We immediately see that this relative distance decreases rapidly with each iteration,which implies that our system has a tendency to distribute heat in such a way thatthe center of heat converges towards the center of mass.

Uniform Distribution

We now switch our attention to the case in which points (xi, yi) are uniformlydistributed over the unit interval. The setup is much the same, where we select a

point at random and go on to distribute heat radially to all points within the fixedradius, r. As before, we plot the center of heat for each iteration, and record itsrelative distance from the center of mass of the system, which is shown as a functionof iteration number, below.

Figure 4: k~rcm ~Rcmk vs. k

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