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Dimitri Lozeve 2018-11-13 22:48:36 +01:00
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<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Dimitri Lozeve - Ising model simulation in APL</title>
<link rel="stylesheet" href="../css/default.css" />
<link rel="stylesheet" href="../css/syntax.css" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" async></script>
</head>
<body>
<header>
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<h1 id="the-ising-model-in-apl">The Ising model in APL</h1>
<p>I needed a small project to try APL while I was learning. Something array-based, obviously. Since I already implemented a Metropolis-Hastings simulation of the <a href="./ising-model.html">Ising model</a>, which is based on a regular lattice, I decided to reimplement it in Dyalog APL.</p>
<p>It is only a few lines long, but I will try to explain what it does step by step.</p>
<p>The first function simply generates a random lattice filled by elements of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>+</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\{-1,+1\}</annotation></semantics></math>.</p>
<p>The first function simply generates a random lattice filled by elements of <span class="math inline">\(\{-1,+1\}\)</span>.</p>
<pre class="apl"><code>L←{(2×?⍵ ⍵⍴2)-3}
</code></pre>
<p>Let’s deconstruct what is done here:</p>
@ -45,7 +47,7 @@
<li>⍵ is the argument of our function.</li>
<li>We generate a ⍵×⍵ matrix filled with 2, using the <code>⍴</code> function: <code>⍵ ⍵⍴2</code></li>
<li><code>?</code> draws a random number between 1 and its argument. We give it our matrix to generate a random matrix of 1 and 2.</li>
<li>We multiply everything by 2 and subtract 3, so that the result is in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>+</mo><mn>1</mn><mo stretchy="false" form="postfix">}</mo></mrow><annotation encoding="application/x-tex">\{-1,+1\}</annotation></semantics></math>.</li>
<li>We multiply everything by 2 and subtract 3, so that the result is in <span class="math inline">\(\{-1,+1\}\)</span>.</li>
<li>Finally, we assign the result to the name <code>L</code>.</li>
</ul>
<p>Sample output:</p>
@ -71,9 +73,9 @@
<li>We extract the x and y coordinates of the site.</li>
<li><code>N</code> is the size of the lattice.</li>
<li><code>xn</code> and <code>yn</code> are respectively the vertical and lateral neighbours of the site. <code>N|</code> takes the coordinates modulo <code>N</code> (so the lattice is actually a torus). (Note: we used <code>⎕IO←0</code> to use 0-based array indexing.)</li>
<li><code>+/</code> sums over all neighbours of the site, and then we multiply by the value of the site itself to get <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Δ</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">\Delta E</annotation></semantics></math>.</li>
<li><code>+/</code> sums over all neighbours of the site, and then we multiply by the value of the site itself to get <span class="math inline">\(\Delta E\)</span>.</li>
</ul>
<p>Sample output, for site <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">(</mo><mn>3</mn><mo>,</mo><mn>3</mn><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">(3, 3)</annotation></semantics></math> in a random <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mo>×</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">5\times 5</annotation></semantics></math> lattice:</p>
<p>Sample output, for site <span class="math inline">\((3, 3)\)</span> in a random <span class="math inline">\(5\times 5\)</span> lattice:</p>
<pre class="apl"><code> 3 3ising.∆E ising.L 5
¯4
</code></pre>
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}
</code></pre>
<ul>
<li>⍺ is the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>β</mi><annotation encoding="application/x-tex">\beta</annotation></semantics></math> parameter of the Ising model, ⍵ is the lattice.</li>
<li>We draw a random site <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math> with the <code>?</code> function.</li>
<li><code>new</code> is the lattice but with the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math> site flipped.</li>
<li>We compute the probability <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mo>exp</mo><mo stretchy="false" form="prefix">(</mo><mo>−</mo><mi>β</mi><mi>Δ</mi><mi>E</mi><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">\alpha = \exp(-\beta\Delta E)</annotation></semantics></math> using the <code>*</code> function (exponential) and our previous <code>∆E</code> function.</li>
<li><code>?0</code> returns a uniform random number in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false" form="prefix">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false" form="postfix">)</mo></mrow><annotation encoding="application/x-tex">[0,1)</annotation></semantics></math>. Based on this value, we decide whether to update the lattice, and we return it.</li>
<li>⍺ is the <span class="math inline">\(\beta\)</span> parameter of the Ising model, ⍵ is the lattice.</li>
<li>We draw a random site <span class="math inline">\((x,y)\)</span> with the <code>?</code> function.</li>
<li><code>new</code> is the lattice but with the <span class="math inline">\((x,y)\)</span> site flipped.</li>
<li>We compute the probability <span class="math inline">\(\alpha = \exp(-\beta\Delta E)\)</span> using the <code>*</code> function (exponential) and our previous <code>∆E</code> function.</li>
<li><code>?0</code> returns a uniform random number in <span class="math inline">\([0,1)\)</span>. Based on this value, we decide whether to update the lattice, and we return it.</li>
</ul>
<p>We can now bring everything together for display:</p>
<pre class="apl"><code>Ising←{' ⌹'[1+1=({10 U ⍵}⍣⍵)L ⍺]}
</code></pre>
<ul>
<li>We draw a random lattice of size ⍺ with <code>L ⍺</code>.</li>
<li>We apply to it our update function, with $<em>β</em>$=10, ⍵ times (using the <code>⍣</code> function, which applies a function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>n</mi><annotation encoding="application/x-tex">n</annotation></semantics></math> times.</li>
<li>We apply to it our update function, with $<em>β</em>$=10, ⍵ times (using the <code>⍣</code> function, which applies a function <span class="math inline">\(n\)</span> times.</li>
<li>Finally, we display -1 as a space and 1 as a domino ⌹.</li>
</ul>
<p>Final output, with a <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>80</mn><mo>×</mo><mn>80</mn></mrow><annotation encoding="application/x-tex">80\times 80</annotation></semantics></math> random lattice, after 50000 update steps:</p>
<p>Final output, with a <span class="math inline">\(80\times 80\)</span> random lattice, after 50000 update steps:</p>
<pre class="apl"><code> 80ising.Ising 50000
⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹ ⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹
⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹ ⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹⌹