Introduction to the Modeling and Analysis of Complex Systems

OutIn232 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGare going through a cross-section of the space per area per unit of time. In this 2-D example,a cross-section means a line segment in the space, and the total area of the sectionis just the length of the line segment.2h × v y (x,y+h)2h ×v x (x-h,y)2hInA small spatial areaaround (x, y)Concentration of “stuff”= # of particles / (2h) 2Out2h ×v x (x+h,y)2h2h × v y (x,y-h)Figure 13.4: Explanation of the physical meaning of divergence in a 2-D space.Then, consider a temporal change of the number of particles, say N, within a verysmall spatial area around (x, y) (Fig. 13.4). The temporal change of N can be calculatedby counting how many particles are coming into/going out of the area through each edgeper unit of time, as∂N∂t = 2hv x(x − h, y) + 2hv y (x, y − h) − 2hv x (x + h, y) − 2hv y (x, y + h), (13.6)where h is the distance from (x, y) to each edge, and therefore the length of each edge is2h.