“Introduction to abstract entropy” by Alex Altair

https://www.lesswrong.com/posts/REA49tL5jsh69X3aM/introduction-to-abstract-entropy#fnrefpi8b39u5hd7

This post, and much of the following sequence, was greatly aided by feedback from the following people (among others): Lawrence Chan, Joanna Morningstar, John Wentworth, Samira Nedungadi, Aysja Johnson, Cody Wild, Jeremy Gillen, Ryan Kidd, Justis Mills and Jonathan Mustin. Illustrations by Anne Ore.

Introduction & motivation

In the course of researching optimization, I decided that I had to really understand what entropy is.[1] But there are a lot of other reasons why the concept is worth studying:

• Information theory:
  • Entropy tells you about the amount of information in something.
  • It tells us how to design optimal communication protocols.
  • It helps us understand strategies for (and limits on) file compression.
• Statistical mechanics:
  • Entropy tells us how macroscopic physical systems act in practice.
  • It gives us the heat equation.
  • We can use it to improve engine efficiency.
  • It tells us how hot things glow, which led to the discovery of quantum mechanics.
• Epistemics (an important application to me and many others on LessWrong):
  • The concept of entropy yields the maximum entropy principle, which is extremely helpful for doing general Bayesian reasoning.
• Entropy tells us how "unlikely" something is and how much we would have to fight against nature to get that outcome (i.e. optimize).
• It can be used to explain the arrow of time.
• It is relevant to the fate of the universe.
• And it's also a fun puzzle to figure out!

I didn't intend to write a post about entropy when I started trying to understand it. But I found the existing resources (textbooks, Wikipedia, science explainers) so poor that it actually seems important to have a better one as a prerequisite for understanding optimization! One failure mode I was running into was that other resources tended only to be concerned about the application of the concept in their particular sub-domain. Here, I try to take on the task of synthesizing the abstract concept of entropy, to show what's so deep and fundamental about it. In future posts, I'll talk about things like: