## What Is Entropy?

#### July 26, 2024

I wrote a little book on entropy:

If you see typos and other mistakes, or have trouble understanding things, please let me know!

An alternative title would be 92 Tweets on Entropy, but people convinced me that title wouldn't age well: in decade or two few people may remember what 'tweets' were.

Here is the foreword, which explains the basic idea.

#### Foreword

Once there was a thing called Twitter, where people exchanged short messages called 'tweets'. While it had its flaws, I came to like it and eventually decided to teach a short course on entropy in the form of tweets. This little book is a slightly expanded version of that course.

It's easy to wax poetic about entropy, but what is it? I claim it's the amount of information we don't know about a situation, which in principle we could learn. But how can we make this idea precise and quantitative? To focus the discussion I decided to tackle a specific puzzle: why does hydrogen gas at room temperature and pressure have an entropy corresponding to about 23 unknown bits of information per molecule? This gave me an excuse to explain these subjects:

• iinformation
• Shannon entropy and Gibbs entropy
• the principle of maximum entropy
• the Boltzmann distribution
• temperature and coolness
• the relation between entropy, expected energy and temperature
• the equipartition theorem
• the partition function
• the relation between expected energy, free energy and entropy
• the entropy of a classical harmonic oscillator
• the entropy of a classical particle in a box
• the entropy of a classical ideal gas.

I have largely avoided the second law of thermodynamics, which says that entropy always increases. While fascinating, this is so problematic that a good explanation would require another book! I have also avoided the role of entropy in biology, black hole physics, etc. Thus, the aspects of entropy most beloved by physics popularizers will not be found here. I also never say that entropy is 'disorder'.

I have tried to say as little as possible about quantum mechanics, to keep the physics prerequisites low. However, Planck's constant shows up in the formulas for the entropy of the three classical systems mentioned above. The reason for this is fascinating: Planck's constant provides a unit of volume in position-momentum space, which is necessary to define the entropy of these systems. Thus, we need a tiny bit of quantum mechanics to get a good approximate formula for the entropy of hydrogen, even if we are trying our best to treat this gas classically.

Since I am a mathematical physicist, this book is full of math. I spend more time trying to make concepts precise and looking into strange counterexamples than an actual 'working' physicist would. If at any point you feel I am sinking into too many technicalities, don't be shy about jumping to the next tweet. The really important stuff is in the boxes. It may help to reach the end before going back and learning all the details. It's up to you.

© 2024 John Baez except animation by Jacopo Bertolotti
baez@math.removethis.ucr.andthis.edu