Editing Stellar dynamics (section)

=== The CBE ===
In plasma physics, the collisionless Boltzmann equation is referred to as the [[Vlasov equation]], which is used to study the time evolution of a plasma's distribution function.

The Boltzmann equation is often written more generally with the [[Liouville operator]] <math>{\mathcal{L}} </math> as
<math display="block">{\mathcal{L}} f(t,\mathbf{x},\mathbf{p}) = {f^\text{Max}_\text{fit} - f(t,\mathbf{x},\mathbf{p}) \over t_\text{relax}},  </math>
<math display="block">
 {\mathcal{L}} \equiv \frac{\partial}{\partial t} + \frac{\mathbf{p}}{m} \cdot \nabla + \mathbf{F}\cdot\frac{\partial}{\partial \mathbf{p}}\,.</math>
where <math>\mathbf{F} \equiv \mathbf{\dot{p}} =-m \nabla \Phi</math> is the gravitational force and <math> f^\text{Max}_\text{fit}</math> is the Maxwell (equipartition) distribution (to fit the same density, same mean and rms velocity as <math> f(t,\mathbf{x},\mathbf{p})</math>).  The equation means the non-Gaussianity will decay on a (relaxation) time scale of <math> t_\text{relax} </math>, and the system will ultimately relaxes to a Maxwell (equipartition) distribution.

Whereas Jeans applied the collisionless Boltzmann equation, along with Poisson's equation, to a system of stars interacting via the long range force of gravity, [[Anatoly Vlasov]] applied Boltzmann's equation with [[Maxwell's equations]] to a system of particles interacting via the [[Coulomb Force]].<ref>{{Cite journal|last=Henon|first=M|date=June 21, 1982|title=Vlasov Equation? |journal=Astronomy and Astrophysics|volume=114|issue=1|pages=211–212|bibcode=1982A&A...114..211H}}</ref> Both approaches separate themselves from the kinetic theory of gases by introducing long-range forces to study the long term evolution of a many particle system. In addition to the Vlasov equation, the concept of [[Landau damping]] in plasmas was applied to gravitational systems by [[Donald Lynden-Bell]] to describe the effects of damping in spherical stellar systems.<ref>{{Cite journal|last=Lynden-Bell |first=Donald|date=1962|title=The stability and vibrations of a gas of stars|journal=Monthly Notices of the Royal Astronomical Society|volume=124|issue=4|pages=279–296|doi=10.1093/mnras/124.4.279|bibcode=1962MNRAS.124..279L|doi-access=free}}</ref>

A nice property of f(t,x,v) is that many other dynamical quantities can be formed by its [[collisionless Boltzmann equation|moments]], e.g., the total mass, local density, pressure, and mean velocity.  Applying the [[collisionless Boltzmann equation]], these moments are then related by various forms of continuity equations, of which most notable are the [[Jeans equations]] and [[Virial theorem]].