The Math of Phase Change - Deriving the Boltzmann Formula and Defining Temperature

Explore the mathematical foundations of phase transitions through a comprehensive 41-minute guest video that derives the Boltzmann formula, defines temperature from first principles, and demonstrates liquid-vapor simulations. Begin by examining what phase change means at a fundamental level, then delve into the Boltzmann law and free energy concepts. Work through preliminary simulations to build intuition before tackling the rigorous definition of temperature and the complete derivation of the Boltzmann law. Learn about Monte Carlo Markov Chain (MCMC) sampling methods and understand the role of chemical potential in phase transitions. Discover fascinating properties of phase change systems, including connections to universal random curves and Schramm-Loewner evolution. Access interactive simulations to experiment with the liquid-vapor model and gain hands-on experience with the theoretical concepts presented. The presentation includes detailed technical notes covering ergodicity in deterministic dynamical systems, discrete versus continuous probability distributions, Glauber dynamics for parallel computation, and renormalization operations for multi-scale analysis.