I did a deep dive on entropy a couple years ago. I found the concept to be much harder to understand than I expected! Specifically, it was confusing to shift from the intuitive but wrong “entropy is disorder” to “entropy is about the number of possible microstates in a macrostate” (Boltzmann Entropy) <a href="https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula" rel="nofollow">https://en.wikipedia.org/wiki/Boltzmann%27s_entropy_formula</a><p>I was extra confused when I discovered that a spread out cloud of hydrogen is lower entropy than the same cloud gravitationally bound together in a star. So entropy isn’t just about “spreading out,” either.<p>I found that Legos provide a really nice example to illustrate entropy, so I’ll share that here.<p>Consider a big pile of Legos, the detritus of many past projects. Intuitively, a pile of Legos is high entropy because it is disordered—but if we are trying to move beyond order/disorder, we need to relate it to micro states and macro states.<p>Therefore, a pile of Legos is high entropy because if you randomly swap positions of the pieces it will all be the same macrostate—ie a big pile of Legos. Nevertheless, each of the Lego pieces is still in a very specific position— and if we could clearly snapshot all those positions, that would be the specific microstate. That means that the macrostate of the pile has an astronomical number of possible microstates — there are many ways to reorganize the pieces that still look like a pile.<p>On the other hand, consider a freshly built Lego Death Star. This is clearly low entropy. But to understand why in terms of microstates, it is because very few Legos can be swapped or moved without it not really being a Death Star anymore. The low entropy is because there are very few microstates (specific Lego positions) that correspond to the given macro state (being a Death Star).<p>This specific case helped me grok Boltzmann entropy. To extend it, consider a box with a small ice crystal in it: this has many fewer possible microstates than the same box filled with steam. In the steam, molecules can pretty much we swapped and moved anywhere and the macrostate is the same. With the crystal, if you start randomly swapping molecules to different microstates, it stops being an ice crystal quickly. So an ice crystal is low entropy.<p>Now, the definition of what counts as a macrostate is very important in this… but this comment is long enough and I still haven’t gotten to the gym…