### Vlasov-Poisson equation

(equation describing plasma ignoring the magnetic field)

The **Vlasov-Poisson equation** is a simplified equation for plasma,
somewhat like the Maxwell-Boltzmann equation is for ideal gases,
but also not presuming equilibrium:
it describes an evolution of an ideal plasma's distribution.
It is a simplified version of the **Vlasov-Maxwell equation**.
These equations model the distribution
of gases of charged particles in which electromagnetic forces are
significant even at a distance.
Essentially, they are the **Boltzmann Transport Equation (BTE)** with no
collision term (the **collisionless Boltzmann equation**,
which assumes the elastic collisions corresponding to
those of neutral gas are insignificant)
and with its force term describing the electromagnetic fields
of that instant due to the moving charged particles.
The *Vlasov-Poisson equation*'s
simplification is to ignore relativity as well as
any effect of magnetic fields.
Either of these equations must be used simultaneously for each
type of charged particle.
In the case of the Vlasov-Poisson equation, they
are used in conjunction with a **Poisson's-equation** description
of the electric field, the whole equation
set termed a **Vlasov-Poisson system**.
All these equations are likely to require numerical analysis
for any real application and simplifications can be useful.

Usage of the term **Vlasov equation** varies: it has been used for
each of these, including merely the *collisionless Boltzmann equation*.
The term **Vlasov simulation** is a simulation, e.g., of plasma,
making use of one of these *Vlasov equations*, generally codes
taking into account Coulomb force, and often ignoring collisions,
which can be insignificant in the thin plasma common in astrophysical
phenomena. The term **Vlasov solver** is also used for such codes,
suggesting the calculation is merely solving equations describing
the process.

(*physics,plasma,mathematics*)
**Further reading:**

https://en.wikipedia.org/wiki/Vlasov_equation

https://en.wikipedia.org/wiki/Boltzmann_equation

https://www.terrapub.co.jp/e-library/amss/pdf/023.pdf

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