Simulating and understanding phase change | Guest video by Vilas Winstein

Deriving the Boltzmann formula, defining temperature, and simulating liquid/vapor. @SpectralCollective has the second part: https://youtu.be/yEcysu5xZH0 You can play with a simulation of this model here: https://vilas.us/simulations/liquidvapor/ These lessons are funded directly by viewers: https://3b1b.co/support Home page: https://www.3blue1brown.com Notes from Vilas: 1) This open problem is to prove the ergodicity of the deterministic dynamical systems that are used to model the molecule-level physics. A good example of such a dynamical system is the box with particles evolving according to Newton's laws with elastic collisions, like in the video. 2) This video assumes that all probability distributions are discrete, which is the case in the simulations. But one can also set up this formalism for systems with continuous state spaces. Again, a good example is particles in a box, which are not restricted to a lattice, which is only used for visualization purposes in this video. 3) Strictly speaking, these "derivatives" don't make sense since in our simulations the energy can only take on a discrete set of values. But a derivative is the right way to think about this, and is the correct notion in a limiting sense. 4) The factor of -T in the definition of the chemical potential is sort of a historical leftover, but including it has the convenient side effect of allowing the same Boltzmann formula to hold, with an energy function that depends on the chemical potential as well. It should also be noted that temperature must equalize in any situation where the chemical potential equalizes, since it is impossible for systems to exchange molecules without also exchanging energy. 5) This algorithm, where we only choose one pixel at a time, is called Glauber dynamics. There are multiple ways to parallelize it, but the method chosen in this video is to only update each pixel with a low probability at each call of the compute shader, to avoid (frequently) updating two nei