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:
