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<!-- Mirrored from www.mathsisfun.com/physics/entropy.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:52:10 GMT -->
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Entropy Introduction</h1>
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<p class="center">Entropy is a measure of disorder</p>
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<p>
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You walk into a room and see a table with coins on it.</p>
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<p class="center"><img src="images/table-coins.svg" alt="table with coins"></p>
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<p>You notice they are all heads up:</p>
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<p class="center large">HHHHHH</p>
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<p>"Whoa, that seems unlikely" you think. But nice and <b>orderly</b>, right?</p>
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<p>You move the table and the vibration causes a coin to flip to tails (T):</p>
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<p class="center large">HHHHTH</p>
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<p>"Huh, I wonder if I can get it to flip back again?", so you shift the table a bit more and get this:</p>
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<p class="center large">HTTHTH</p>
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<p>Hmmm... more disorderly. You shift the table a bit more and still get random heads and tails.</p>
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<p>To begin with they were very orderly, but now they are disorderly again and again.</p>
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<p>We can <b>see </b>they are disorderly, but can we come up with a <b>measure </b>of how disorderly they are?</p>
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<p>First, how many possible states can they be in?</p>
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<ul>
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<li>1 coin can have 2 states: {H, T} </li>
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<li>2 coins can have 4 states: {HH, HT, TH, TT} </li>
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<li>3 coins can have 8 states: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} </li></ul>
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<p>They double each time, so 6 coins can have 2<sup>6</sup> = <b>64 states</b></p>
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<p>Each state has exactly the same chance, but let us <b>group </b>them by how many tails:</p>
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<div class="simple center">
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<table>
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<tbody>
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<tr>
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<th>Tails</th>
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<th>States</th>
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<th>Count of States</th></tr>
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<tr>
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<td>0</td>
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<td>HHHHHH</td>
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<td>1</td></tr>
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<tr>
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<td>1</td>
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<td>HHHHHT HHHHTH HHHTHH HHTHHH HTHHHH THHHHH</td>
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<td>6</td></tr>
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<tr>
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<td>2</td>
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<td>HHHHTT HHHTHT HHHTTH HHTHHT HHTHTH HHTTHH HTHHHT HTHHTH HTHTHH HTTHHH THHHHT THHHTH THHTHH THTHHH TTHHHH</td>
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<td>15</td></tr>
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<tr>
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<td>3</td>
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<td>HHHTTT HHTHTT HHTTHT HHTTTH HTHHTT HTHTHT HTHTTH HTTHHT HTTHTH HTTTHH THHHTT THHTHT THHTTH THTHHT THTHTH THTTHH TTHHHT TTHHTH TTHTHH TTTHHH</td>
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<td>20</td></tr>
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<tr>
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<td>4</td>
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<td>HHTTTT HTHTTT HTTHTT HTTTHT HTTTTH THHTTT THTHTT THTTHT THTTTH TTHHTT TTHTHT TTHTTH TTTHHT TTTHTH TTTTHH</td>
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<td>15</td></tr>
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<tr>
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<td>5</td>
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<td>HTTTTT THTTTT TTHTTT TTTHTT TTTTHT TTTTTH</td>
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<td>6</td></tr>
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<tr>
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<td>6</td>
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<td>TTTTTT</td>
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<td>1</td></tr></tbody></table>
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</div>
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<p>Only 1 of the 64 possibilities is HHHHHH. </p>
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<p>A combination of H and T is much more likely.</p>
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<p>The counts (1, 6, 15, 20, 15, 6, 1) give a <b>rough </b>idea of disorder, but we can do better! </p>
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<p>It turns out that a <a href="../algebra/logarithms.html">logarithm</a> of the <b>number of states</b> is perfect for disorder.</p>
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<p>Here we use the natural logarithm "ln" to 2 decimal places: </p>
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<div class="simple center">
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<table class="center">
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<tbody>
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<tr>
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<th>Tails</th>
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<th>States</th>
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<th><i><b>ln</b></i>(States)</th></tr>
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<tr>
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<td>0</td>
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<td>1<br>
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</td>
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<td>0</td></tr>
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<tr>
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<td>1</td>
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<td>6<br>
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</td>
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<td>1.79</td></tr>
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<tr>
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<td>2</td>
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<td>15<br>
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</td>
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<td>2.71</td></tr>
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<tr>
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<td>3</td>
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<td>20<br>
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</td>
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<td>3.00</td></tr>
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<tr>
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<td>4</td>
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<td>15<br>
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</td>
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<td>2.71</td></tr>
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<tr>
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<td>5</td>
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<td>6<br>
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</td>
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<td>1.79</td></tr>
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<tr>
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<td>6</td>
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<td>1<br>
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</td>
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<td>0</td></tr></tbody></table>
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</div>
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<p>And that is Entropy! We throw in a constant "k" and get:</p>
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<p class="center large">Entropy = k <i><b>ln</b></i>(States)</p>
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<p>Play with it here. Every time a random spot is chosen to be flipped. Are any lines more common? Are any totals more common?</p>
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<div class="script" style="height: 330px;">
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images/entropy.js
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</div>
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<p>This concept helps explain many things in the world: milk mixing in coffee, movement of heat, pollution, gas dispersing and more.</p>
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<p>In the real world there are <b>many </b>more particles, and each particle has <b>many</b> more states, but the same idea applies.</p>
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<h2>Gas</h2>
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<p>Here is a balloon of gas in a plastic box:</p>
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<p class="center"><img src="images/gas-box-stt.svg" alt="gas in balloon in box"></p>
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<p>The gas molecules bounce around inside the balloon in different directions at different speeds. </p>
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<p>There are <b>many </b>different states the gas can be in. </p>
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<p class="center large">By "many" we mean a very very very large number.</p>
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<p>The balloon bursts and the gas spreads out into the box.</p>
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<p class="center"><img src="images/gas-box-all.svg" alt="gas in balloon in box"></p>
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<p></p>
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<p>Now there are <b>many more </b>possible states:</p>
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<ul>
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<li>some of those states have the gas back to the balloon shape again (not likely!)</li>
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<li>other states might form the word "HI" (not likely!)</li>
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<li>but <b>most of the new states</b> are going to be well spread out in the space available.</li></ul>
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<p>So the new states include the old states plus many many more.</p>
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<p>The value of <span class="large"><i><b>ln</b></i>(States)</span> is now <b>larger</b>, so entropy has <b>increased</b>.</p>
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<p class="center larger">As a general rule <b>entropy increases</b>. </p>
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<p> But let us be clear here:</p>
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<p>Any <b>one </b>state (imagine we froze time) is just as likely as any other state. </p>
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<p>But entropy is about a <b>group </b>or class of states. </p>
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<ul>
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<li>"All states inside the balloon"</li>
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<li>"All states within the whole box"</li></ul>
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<p>Similar to the example at the start:</p>
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<ul>
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<li>"Only 1 Head"</li>
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<li>"2 Heads"</li>
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<li>"3 Heads"</li>
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<li>etc...</li></ul>
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<p>In picture form:</p>
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<table class="center" style="text-align:center;">
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<tbody>
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<tr>
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<td><img src="images/gas-box-state2.svg" alt="gas in box 2"><br>1 State</td>
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<td> </td>
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<td><img src="images/gas-box-state1.svg" alt="gas in box 1"><br>1 State</td></tr>
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<tr>
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<td rowspan="1" colspan="3" class="larger">Any 1 state is equally likely</td></tr>
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</tbody></table>
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<p><br></p>
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<table class="center" style="text-align:center;">
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<tbody>
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<tr>
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<td><img src="images/gas-box-all-ball.svg" alt="gas in balloon in box"><br>Many States</td>
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<td> </td>
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<td><img src="images/gas-box-all.svg" alt="gas in balloon in box"><br>Many <b>Many </b>States</td></tr>
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<tr>
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<td rowspan="1" colspan="3" class="larger">But groups can have very different numbers of states</td></tr>
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</tbody></table>
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<div class="words">
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<p>Any single state is called a "microstate". Each are equally likely, no matter how weird they may look.</p>
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<p>The groups are called "macrostates", and because they may contain different numbers of microstates they are not equally likely.</p>
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</div>
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<h2>Entropy Increases</h2>
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<p>With just 6 coins we saw entropy naturally increase, but with some chance of getting lower entropy (HHHHHH has a 1/64 chance)</p>
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<p>Now imagine 100 coins: the chance of all heads is less than 0.000 000 000 000 000 000 000 000 000 001, which would be freaky.</p>
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<p>Now imagine a drop of water with over 5 x 10<sup>21</sup> atoms (and an atom is more complex than heads or tails). The chance of randomly getting reduced entropy is so ridiculously small that we just say <b>entropy increases</b>.</p>
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<p>And this is the main idea behind the <b>Second Law of Thermodynamics</b>. </p>
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<h2>Entropy Decreases</h2>
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<p>Ah, but we <b>can </b>make entropy decrease <b>in a region</b>, but at the expense of increasing entropy elsewhere. </p>
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<p>Examples:</p>
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<ul>
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<li>A factory that makes neat stacks of paper. The paper is orderly but the factory creates a lot of disorder to make them.</li>
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<li>A new building is neat and orderly, but making it created a lot of disorder (quarries, timber mills, steel production, electricity, fuel, etc)</li></ul>
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<h2>Physics</h2>
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<p>Entropy behaves in predictable ways.</p>
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<p>In Physics the basic definition is:</p>
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<p class="center large">S = k<sub>B</sub> log(Ω)</p>
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<p>Where:</p>
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<ul>
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<li>S is entropy</li>
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<li>k<sub>B</sub> is Boltzmann's Constant</li>
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<li>Ω is the number of "Microstates"</li></ul>
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<p>Another important formula is:</p>
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<p class="center large">ΔS = <span class="intbl"><em>Q</em><strong>T</strong></span></p>
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<!-- DELS = Q/T -->
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<p>Where:</p>
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<ul>
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<li>ΔS is the change in entropy</li>
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<li>Q is the flow of heat energy in or out of the system</li>
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<li>T is temperature</li></ul>
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<p>But more details are beyond this introductory page.</p>
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<p><br></p>
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<div class="def">
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<h3>Footnote: Logarithm Bases</h3>
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<p>We used the natural logarithm because we love it. Other people prefer base 2 or base 10 logarithms. Any base is fine because we can convert between them using constants such as ln(2) or ln(10) like this:</p>
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<ul>
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<li>log<sub>2</sub>(15) = ln(15)/ln(2)</li>
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<li>log<sub>10</sub>(15) = ln(15)/ln(10) </li></ul>
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</div><p><br>
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</p>
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