2b:null 2c:["$","title",null,{"children":"Critical states | Chuncheng | Observable"}] 31:[] 2f:{"id":"484f573366f2f1ca","slug":"critical-states","title":"Critical states","thumbnail":"9aef6b2bf889a924bf0683e39626dd268a0162d46919190f9769e33e60b08746","access_level":"public","create_time":"$D2024-09-28T06:14:14.577Z","update_time":"$D2024-09-28T10:05:44.612Z","view_time":null,"first_public_time":"$D2024-09-28T06:14:22.899Z","fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"bcd65cef1cb4d2ed","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":"starter_2024"},"version":342,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"16328058352682a8","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":null}],"members":[{"type":"individual","id":"16328058352682a8","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":null,"role":"editor"},{"type":"team","id":"bcd65cef1cb4d2ed","login":"listenzcc","name":"Chuncheng","avatar_url":"https://avatars-next.observableusercontent.com/bb3c68e77a708a587ffc2d663bb83421f20d67991dbe6c1623c6e5d2fb73d15e","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"fec57c7fdd1e4c68","slug":"critical-state","title":"critical state","owner":{"id":"bcd65cef1cb4d2ed","login":"listenzcc","name":"Chuncheng"}},"type":"notebook","body":{"title":"Critical states","cells":[{"id":0,"value":"# Critical states","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"---\n## Boltzmann distribution\nwiki: \n\nIn statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution[1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form:\n\n${tex`p_i \\propto \\exp(-\\frac{\\varepsilon_i}{kT})`}\n\nwhere, a constant kT of the distribution is the product of the Boltzmann constant k and thermodynamic temperature T.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":29,"value":"---\n## Interactive graphics","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":32,"value":"viewof kT = Inputs.range([0.1, 5], { label: \"kT\", step: 0.1, value: 3.2 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"viewof toggle = Inputs.toggle({ label: \"Contour or Value?\", value: true })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":322,"value":"

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":315,"value":"{\n while (true) {\n yield draw(new Date().getTime() % 10000);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"draw = (n = \"Now\") => {\n let plt = Plot.plot({\n title: `Heatmap: ${n}`,\n x: { nice: true, axis: false },\n y: { nice: true, axis: false, reverse: toggle },\n\n color: { scheme: \"Blues\" },\n aspectRatio: 1.0,\n marks: [\n toggle\n ? Plot.contour(data, {\n x: \"x\",\n y: \"y\",\n stroke: \"c\",\n blur: 5,\n interval: 3\n })\n : Plot.cell(data, {\n x: \"x\",\n y: \"y\",\n fill: \"c\",\n strokeWidth: 1,\n stroke: \"c\"\n })\n ]\n });\n\n\n\n return plt;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":294,"value":"{\n let rnd = d3.randomUniform(),\n kTInv = 1 / kT;\n\n data.map((d) => {\n if (rnd() < 0.0) {\n return;\n }\n\n let { v, neighbors } = d,\n u = 1 - v,\n e0 = d3.sum(neighbors, (i) => Math.abs(data[i].v - v)),\n e1 = d3.sum(neighbors, (i) => Math.abs(data[i].v - u)),\n r = Math.exp((e1 - e0) * kTInv),\n p = r / (1 + r),\n w = rnd() < p ? v : u;\n\n d.w = w;\n });\n\n data.map((d) => {\n d.v = d.w;\n d.c = d.v + d3.sum(d.neighbors, (i) => data[i].v);\n });\n\n return now;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"Plot.plot({\n title: \"Boltzmann curves shape\",\n x: { nice: true },\n y: { nice: true },\n color: { legend: true, scheme: \"Blues\", range: [0.8, 0.1] },\n grid: true,\n marks: [Plot.line(boltzmannDistribution, { x: \"ei\", y: \"p\", stroke: \"kT\" })]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"boltzmannDistribution = {\n let data = [],\n p;\n\n for (let t = 1; t < 100; ++t) {\n for (let i = 0; i < 5; i += 0.1) {\n p = Math.exp(-i / t);\n data.push({ kT: t, ei: i, p: p });\n }\n }\n\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"data = {\n let width = 100,\n height = 100,\n rnd = d3.randomInt(2),\n data = [];\n\n for (let i = 0; i < width; ++i) {\n for (let j = 0; j < height; ++j) {\n data.push({ x: i, y: j, neighbors: [], v: rnd() });\n }\n }\n\n data.map((d, offset) => {\n let { x, y, neighbors, v } = d;\n\n for (let i = 0; i < 3; ++i) {\n for (let j = 0; j < 3; ++j) {\n if ((i === 1) & (j === 1)) {\n continue;\n }\n neighbors.push(\n data.findIndex((d) => (d.x === x + i - 1) & (d.y === y + j - 1))\n );\n }\n }\n\n // neighbors.push(data.findIndex((d) => (d.x === x - 1) & (d.y === y)));\n // neighbors.push(data.findIndex((d) => (d.x === x + 1) & (d.y === y)));\n // neighbors.push(data.findIndex((d) => (d.x === x) & (d.y === y - 1)));\n // neighbors.push(data.findIndex((d) => (d.x === x) & (d.y === y + 1)));\n\n d.neighbors = neighbors.filter((d) => d > -1);\n d.w = v;\n d.c = v;\n });\n\n return data;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":21,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"d3 = require(\"d3\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 30:"$2f:body"