---
tag: SJH7M2VCTE
kind: concept
name: Vlasov equation
msc: [35Q83, 82C40]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Vlasov_equation
same_as:
  wikidata: Q2323892
  wikipedia: Vlasov equation
---

## Summary

The Vlasov equation describes a large system of particles that interact through long-range forces.
Examples are charged particles in a plasma and stars in a galaxy.
It is a [[transport equation|PVD7VZJZ9B]] in position and velocity for the particle density $f(t, x, v)$, without a collision term.
The force is the mean field created by all particles together, so the equation is nonlinear.
Coupled with the Poisson equation it gives the Vlasov–Poisson system.

## Explanations

### L1 human

In a plasma each electron feels the electric pull and push of billions of other charges.
Close encounters between two particles matter little, but the combined field of all of them matters a lot.
The Vlasov equation follows the cloud of particles in the space of positions and velocities, as each particle moves under this averaged force.
For a single particle in the force $-x$ on a line, the orbits in the $(x, v)$ plane are circles.
Any density that is constant on these circles stays the same over time.