2a:null 2b:["$","title",null,{"children":"Heat Capacity of Ising Model | Fedor Indutny | Observable"}] 30:T60d,liveData = { const data = []; let globalSamples = 0; let temperature = MAX_TEMPERATURE; for (;;) { temperature -= TEMPERATURE_STEP; if (temperature < MIN_TEMPERATURE) { temperature += MAX_TEMPERATURE - MIN_TEMPERATURE - Math.random() * TEMPERATURE_STEP; } const beta = 1 / (temperature + 1e-23); let steps = 0; const samples = []; for (let s = 0; s < SAMPLES; s++, globalSamples++) { if (globalSamples % REINIT_EVERY === 0) { initialize(); // Warm-up for (let i = 0; i < WARM_UP_ITERATIONS; i++) { for (let j = 0; j < ITERATION_STEPS; j++, steps++) { metropolisStep(beta, steps); if (steps % REDRAW_EVERY === 0) { draw(); yield data; } } } } const block = []; for (let i = 0; i < BLOCK_SIZE; i++) { for (let j = 0; j < ITERATION_STEPS; j++, steps++) { metropolisStep(beta, steps); if (steps % REDRAW_EVERY === 0) { draw(); yield data; } } block.push(computeEnergy()); } samples.push(computeStatistics(block).mean); } const energyVariance = computeStatistics(samples).variance; const capacity = energyVariance * (beta ** 2); data.push({ Temperature: temperature, 'Heat Capacity': capacity }); if (data.length > MAX_SCATTER_POINTS) { break; } // Re-render plot yield data; } return data; }31:[] 2e:{"id":"d6ba3495f6e11d07","slug":"heat-capacity-of-ising-model","title":"Heat Capacity of Ising Model","thumbnail":"3087b87387477c4fe200ca60aa7df7ea08540b4d645ce2f47a7a2cfcf75bf43b","access_level":"public","create_time":"$D2020-05-26T16:00:09.030Z","update_time":"$D2020-10-06T18:19:19.632Z","view_time":null,"first_public_time":"$D2020-05-26T17:21:05.111Z","fork":false,"stars":3,"starred":false,"owner":{"type":"team","id":"45d16fa014230070","login":"indutny","name":"Fedor Indutny","avatar_url":"https://avatars-next.observableusercontent.com/f790db7ff83b431795ea6c50023d688d196b79183c11c2b18432110463f6d4a3","tier":"starter_2024"},"version":886,"publish_version":886,"role":null,"authored":false,"authors":[{"type":"individual","id":"d26938fddbf9d7bf","login":"indutny","name":"Fedor Indutny","avatar_url":"https://avatars-next.observableusercontent.com/f790db7ff83b431795ea6c50023d688d196b79183c11c2b18432110463f6d4a3","tier":null}],"members":[{"type":"individual","id":"d26938fddbf9d7bf","login":"indutny","name":"Fedor Indutny","avatar_url":"https://avatars-next.observableusercontent.com/f790db7ff83b431795ea6c50023d688d196b79183c11c2b18432110463f6d4a3","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"title":"Heat Capacity of Ising Model","cells":[{"id":0,"value":"md`# Heat Capacity of Ising Model`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":221,"value":"vl.markLine()\n .data(liveData)\n .encode(\n vl.x().field('Temperature').type('quantitative'),\n vl.y().field('Heat Capacity').type('quantitative')\n )\n .render()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":14,"value":"canvas = {\n const canvas = document.createElement('canvas');\n const ctx = canvas.getContext('2d');\n canvas.width = FIELD_WIDTH;\n canvas.height = FIELD_HEIGHT;\n ctx.fillStyle = 'transparent';\n ctx.fillRect(0,0, canvas.width, canvas.height);\n \n return canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":74,"value":"FIELD_WIDTH = Math.min(width, 5)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":80,"value":"FIELD_HEIGHT = FIELD_WIDTH","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":92,"value":"INTERACTION = 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":290,"value":"EXTERNAL_FIELD = 0.0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":155,"value":"ITERATION_STEPS = FIELD_WIDTH * FIELD_HEIGHT","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":210,"value":"WARM_UP_ITERATIONS = 1000","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":189,"value":"MAX_SCATTER_POINTS = 100000","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":322,"value":"MIN_TEMPERATURE = 0.5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":316,"value":"MAX_TEMPERATURE = 5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":319,"value":"TEMPERATURE_STEP = 0.05","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":175,"value":"SAMPLES = 1024","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":704,"value":"REINIT_EVERY = 100000","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":506,"value":"BLOCK_SIZE = 20","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":478,"value":"REDRAW_EVERY = 128 * 1024","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":107,"value":"context = canvas.getContext('2d')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":116,"value":"imageData = context.createImageData(FIELD_WIDTH, FIELD_HEIGHT)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":39,"value":"rgbaField = imageData.data","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":810,"value":"neighborField = new Int8Array(FIELD_WIDTH * FIELD_HEIGHT)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":575,"value":"CHANNELS = 4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":579,"value":"ALPHA_CHANNEL = 3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":43,"value":"function initialize() {\n rgbaField.fill(0);\n for (let i = 0; i < rgbaField.length; i += CHANNELS) {\n const value = Math.random() > 0.5 ? 255 : 0;\n \n rgbaField[i + ALPHA_CHANNEL] = value;\n }\n \n for (let y = 0; y < FIELD_HEIGHT; y++) {\n for (let x = 0; x < FIELD_WIDTH; x++) {\n neighborField[y * FIELD_WIDTH + x] = sumNeighbors(x, y);\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":105,"value":"function draw() {\n context.fillRect(0, 0, FIELD_HEIGHT, FIELD_HEIGHT);\n context.putImageData(imageData, 0, 0);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":71,"value":"function sumNeighbors(x, y) {\n let top = (y + FIELD_HEIGHT - 1) % FIELD_HEIGHT;\n let bottom = (y + 1) % FIELD_HEIGHT;\n let left = ((x + FIELD_WIDTH - 1) % FIELD_WIDTH);\n let right = ((x + 1) % FIELD_WIDTH);\n \n // Scale for RGBA lookup\n top *= FIELD_WIDTH * CHANNELS;\n bottom *= FIELD_WIDTH * CHANNELS;\n left *= CHANNELS;\n right *= CHANNELS;\n \n const topLeft = rgbaField[top + left + ALPHA_CHANNEL] >>> 7;\n const topRight = rgbaField[top + right + ALPHA_CHANNEL] >>> 7;\n const bottomLeft = rgbaField[bottom + left + ALPHA_CHANNEL] >>> 7;\n const bottomRight = rgbaField[bottom + right + ALPHA_CHANNEL] >>> 7;\n \n let adjacent = topLeft + topRight + bottomLeft + bottomRight;\n \n // Transform from {0, 1} to {-1, -1}\n adjacent = adjacent * 2 - 4;\n return adjacent;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":814,"value":"function updateNeighbors(x, y, delta) {\n let top = (y + FIELD_HEIGHT - 1) % FIELD_HEIGHT;\n let bottom = (y + 1) % FIELD_HEIGHT;\n let left = ((x + FIELD_WIDTH - 1) % FIELD_WIDTH);\n let right = ((x + 1) % FIELD_WIDTH);\n \n top *= FIELD_WIDTH;\n bottom *= FIELD_WIDTH;\n \n neighborField[top + left] += delta;\n neighborField[top + right] += delta;\n neighborField[bottom + left] += delta;\n neighborField[bottom + right] += delta;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":63,"value":"function metropolisStep(beta, steps) {\n let x = (Math.random() * FIELD_WIDTH) | 0;\n let y = (Math.random() * FIELD_HEIGHT) | 0;\n \n x %= FIELD_WIDTH;\n y %= FIELD_HEIGHT;\n \n const cellOffset = (y * FIELD_WIDTH + x) * CHANNELS + ALPHA_CHANNEL;\n const spin = (rgbaField[cellOffset] >>> 7) * 2 - 1;\n const newSpin = -spin;\n\n const neighbors = neighborField[y * FIELD_HEIGHT + x];\n const energyDelta = -(newSpin - spin) * (neighbors * INTERACTION + EXTERNAL_FIELD);\n\n const acceptance = Math.exp(-beta * energyDelta);\n const u = Math.random();\n if (acceptance >= u) {\n const newCell = (((newSpin + 1) / 2) * 255) | 0;\n rgbaField[cellOffset] = newCell;\n updateNeighbors(x, y, newSpin - spin);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":185,"value":"function computeEnergy() {\n let result = 0;\n for (let x = 0; x < FIELD_WIDTH; x++) {\n for (let y = 0; y < FIELD_HEIGHT; y++) {\n const spin = (rgbaField[(y * FIELD_WIDTH + x) * 4 + 3] >>> 7) * 2 - 1;\n result += -spin * (sumNeighbors(x, y) * INTERACTION + EXTERNAL_FIELD);\n }\n }\n \n // Normalize\n return result / FIELD_WIDTH / FIELD_HEIGHT;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":488,"value":"function meanSpin() {\n let res = 0;\n for (let i = 3; i < rgbaField.length; i += 4) {\n res += (rgbaField[i] >>> 7) * 2 - 1;\n }\n return res / FIELD_WIDTH / FIELD_HEIGHT;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":711,"value":"function computeStatistics(list, start = 0, end = list.length) {\n let mean = 0;\n let variance = 0;\n\n for (let i = start; i < end; i++) {\n const item = list[i];\n\n mean += item;\n variance += item ** 2;\n }\n\n mean /= (end - start);\n variance /= (end - start);\n\n variance -= mean ** 2;\n\n return { mean, variance };\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":126,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":4,"value":"import { vl } from \"@vega/vega-lite-api\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W31","stdlib":"1","titling":"auto"}} 2f:"$2e:body"