---
tag: MH6CSWD7F0
kind: concept
name: Boltzmann equation
msc: [35Q20, 82C40]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Boltzmann_equation
same_as:
  wikidata: Q891653
  wikipedia: Boltzmann equation
---

## Summary

The Boltzmann equation describes a dilute gas through the density $f(t, x, v)$ of particles at position $x$ with velocity $v$.
Particles move freely in straight lines and change velocity only in binary collisions, which the collision operator accounts for.
It links the motion of individual molecules with the equations of fluid dynamics.
Boltzmann's H-theorem shows that a certain entropy cannot decrease, which explains the approach to equilibrium.

## Explanations

### L1 human

A gas consists of a huge number of molecules flying around and bumping into each other.
Instead of following each molecule, one keeps track of how many molecules are near each point with each velocity.
Between collisions this distribution just drifts along with the molecules.
Collisions reshuffle velocities but keep the total momentum and energy.
Over time the gas settles into the Maxwell distribution, the bell-shaped spread of velocities of a gas at rest at a fixed temperature.