Dynamics and Entropy

How much new information can a deterministic system reveal? We start with Hamilton's equations and follow a patch of phase space as it stretches, folds, and winds around—without changing its area. That geometric invariant leads us to stationary processes and a way to measure how unpredictable a system remains after we know its past: dynamical entropy. Along the way, we'll see why a book compresses better than its scrambled letters, what changes when successive events are correlated, and how entropy can show that one dynamical system cannot simulate another. At the end, we'll use entropy to understand the probability of typical long histories. This also sets up the next video, on an application of entropy to number theory through work of Terence Tao. Chapters: 00:00 A secret invariant 00:11 From Newton to Hamilton 00:41 Potential energy and force 01:26 Hamilton's equations 02:47 The pendulum in phase space 03:52 Beyond the pendulum 05:46 Flowing a box: area stays the same 07:08 Four-dimensional volume 07:34 Uncertainty in phase space 08:59 Invariant probability 09:43 From dynamics to random variables 10:38 Stationary processes 12:04 Entropy and conditional entropy 13:12 Dividing phase space into cells 14:32 The entropy rate 15:41 What entropy says about simulation 16:03 Compressing dependent sequences 17:03 Language, DNA, and independence 17:56 The probability of an exact outcome 18:27 Typical long sequences 19:09 What's next 19:33 Favorite flows Animated with Manim Community Edition. I used Codex for help programming but the script, storyboard and voice are all me. Music: “Atmosphere Pulse” by leberch, via Pixabay. https://pixabay.com/music/pulses-atmo... Breaking-glass footage: “Breaking glass with sound” by Trevor Cox, CC BY 3.0. Excerpted, resized, accelerated, and combined with new audio. https://commons.wikimedia.org/wiki/Fi... https://creativecommons.org/licenses/... Earth imagery: NASA Earth Observatory's Blue Marble. https://visibleearth.nasa.gov/images/... Moon color and elevation data: NASA Scientific Visualization Studio's CGI Moon Kit (visualizer Ernie Wright). https://svs.gsfc.nasa.gov/4720/ Books in the compression comparison: public-domain texts from Project Gutenberg by Lewis Carroll, Mary Shelley, Robert Louis Stevenson, Arthur Conan Doyle, Jane Austen, Charles Dickens, Herman Melville, and George Eliot. https://www.gutenberg.org/ The code montage features Google Brotli and Zopfli, Xiph Opus, and Meta Zstandard. Quarter photographs: Numismatic Guaranty Corporation, via Wikimedia Commons.