This is a pure optimization problem. We will center our unit square at the origin, so the particles are located at $(x_i,y_i)$, with $-\\tfrac{1}{2}\\leq x_i\\leq \\tfrac{1}{2}$ and $-\\tfrac{1}{2}\\leq y_i\\leq \\tfrac{1}{2}$. Our optimization problem is rather simple to state:
\n\\begin{align*}
\n\\underset{x,y}{\\text{minimize}}\\qquad & \\sum_{1\\leq i \\lt j \\leq n} \\frac{1}{\\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}} \\\\
\n\\text{subject to:}\\qquad & -\\tfrac{1}{2}\\leq x_i\\leq \\tfrac{1}{2} \\quad i=1,\\dots,n\\\\
\n& -\\tfrac{1}{2}\\leq y_i\\leq \\tfrac{1}{2} \\quad i=1,\\dots,n
\n\\end{align*}Taking a numerical approach, I used a local gradient-based solver in Mathematica with randomized initial conditions, solved 50 times, and took the best solution found. I solved the problem for $n=2,\\dots,17$ and here were the best solutions found:<\/p>\n

The case $n=9$<\/h3>\n
For the case $n=9$, interestingly, the optimal solution is not<\/em> to place the points in a regular $3\\times 3$ grid. Using an irregular arrangement can produce a slightly lower cost, as shown in the figure below.<\/p>\n
$\"\"$ <\/a><\/p>\n
The optimal arrangement for $9$ particles shown above has the particles at coordinates:
\n\\[
\n\\left(
\n\\begin{array}{cc}
\n -0.5 & 0.5 \\\\
\n 0.5 & 0.5 \\\\
\n 0.5 & -0.5 \\\\
\n -0.5 & -0.5 \\\\
\n 0. & 0.5 \\\\
\n 0.5 & 0.0469081 \\\\
\n -0.5 & 0.0469081 \\\\
\n 0.181044 & -0.5 \\\\
\n -0.181044 & -0.5 \\\\
\n\\end{array}
\n\\right)
\n\\]<\/p>\n
Side note:<\/strong> If we change the energy function to $1\/r^2$, then the optimal arrangement does become the regular $3\\times 3$ grid.<\/p>\n<\/div>\n
<\/p>\n<\/body>","protected":false},"excerpt":{"rendered":"
This week\u2019s Fiddler is an optimization problem about fitting particles in a box. You have three particles inside a unit square that all repel one another. The energy between each pair of particles is $1\/r$, where $r$ is the distance between them. To be clear, the particles can be anywhere inside the square or on … Continue reading “Particles in a box”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":4162,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"categories":[42],"tags":[43,10,26],"class_list":["post-4158","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-the-fiddler","tag-fiddler","tag-geometry","tag-optimization"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t