Phase Transitions in a Discretized Fluid: Boltzmann Law and Kawasaki Dynamics

Overview

3Blue1Brown presents a visually intuitive exploration of phase transitions using a discretized liquid vapor model on a 2D lattice. Blue pixels denote molecules and white pixels empty space. Temperature and chemical potential control the system's equilibrium behavior, producing gas or droplet like liquid macrostates. The narrative centers on the Boltzmann distribution, entropy, and free energy as the guiding principles for equilibrium. The sampling method Kawasaki dynamics, a Markov chain Monte Carlo procedure, demonstrates how the system evolves toward the Boltzmann distribution by local random swaps. The result is a two parameter phase diagram that mirrors real world fluids, including a supercritical region and a line separating gas and liquid phases.

Introduction and Motivation

The video introduces a minimal yet powerful discretized fluid model to illustrate phase transitions. By mapping molecules to blue pixels and vacancies to white pixels on a fixed size grid, the talk shows how the system behaves differently as the temperature and a chemical potential parameter are varied. Temperature controls the energy term, while the chemical potential effectively tunes the number of molecules in the box. At high temperatures the system behaves like a gas with high entropy, and at low temperatures molecules tend to cluster, forming droplets like a liquid. This competition between energy minimization and entropy maximization underpins the emergence of distinct macrostates in a way that resembles real water in its phase behavior.

Boltzmann Distribution and Free Energy

The Boltzmann distribution assigns probability to microstates proportionally to exp(-E/T). The probability weight of all microstates with a given energy E scales with the number of microstates at that energy, captured by entropy S. The most probable energy minimizes F = E - T S, the free energy. When T is small, energy dominates and low energy configurations prevail; when T is large, entropy dominates and high entropy configurations prevail. This framework explains why droplets form at low temperature and why high temperature favors dispersed gas like states. The model emphasizes the energetic benefit of neighboring molecule interactions, encoded as a negative energy for adjacent pairs.

Sampling with Kawasaki Dynamics

Enumerating all microstates is impractical, so the video uses Kawasaki dynamics to sample from the Boltzmann distribution. At each step two pixels are chosen, and a swap is accepted with a probability determined by the energy difference and the temperature. This local update depends only on the neighboring pixels, making the algorithm efficient and GPU friendly. When allowing the number of molecules to fluctuate via a chemical potential, the energy function becomes E minus mu times N, leading to a richer two dimensional phase diagram in temperature and chemical potential. The sampling scheme is a Markov chain, ensuring convergence to the equilibrium distribution given enough steps.

Phase Diagram and Macrostates

The resulting phase diagram displays a gas region at high temperature, a liquid region at low temperature, and a smoothly varying supercritical region where the density changes continuously with the control parameters. Droplets emerge as temperature crosses the phase boundary, and their shapes reflect the underlying lattice. The talk also discusses meta stability, where droplets might fail to nucleate and grow without an artificial seed, a phenomenon observed in real systems as well. The Ising model is introduced as a magnetic analogue, and the XY model with vortices demonstrates how changing dimensionality and order parameters yield different critical phenomena. universality is highlighted as a key idea: many details of microscopic interactions do not alter the qualitative macroscopic behavior.

Outlook

The video foreshadows a second part that reduces the model to an even more tractable form, enabling rigorous mathematical understanding of phase transitions using basic tools. The broader message is that simple microscopic rules can generate the rich phenomenology of phase transitions and that universality binds diverse systems under a shared thermodynamic framework.