Editing Stellar dynamics (section)

== Connections to statistical mechanics and plasma physics ==
The statistical nature of stellar dynamics originates from the application of the [[kinetic theory of gases]] to stellar systems by physicists such as [[James Jeans]] in the early 20th century. The [[Jeans equations]], which describe the time evolution of a system of stars in a gravitational field, are analogous to [[Euler equations (fluid dynamics)|Euler's equations]] for an ideal fluid, and were derived from the [[collisionless Boltzmann equation]]. This was originally developed by [[Ludwig Boltzmann]] to describe the non-equilibrium behavior of a thermodynamic system. Similarly to statistical mechanics, stellar dynamics make use of distribution functions that encapsulate the information of a stellar system in a probabilistic manner. The single particle phase-space distribution function, <math>f(\mathbf{x},\mathbf{v},t)</math>, is defined in a way such that
<math display="block">f(\mathbf{x},\mathbf{v},t) \, d\mathbf{x} \, d\mathbf{v} = dN </math>
where <math> dN/N </math> represents the probability of finding a given star with position <math>\mathbf{x}</math> around a differential volume <math>d\mathbf{x}</math> and velocity <math>\text{v}</math> around a differential velocity space volume <math>d\mathbf{v}</math>. The distribution function is normalized (sometimes) such that integrating it over all positions and velocities will equal N, the total number of bodies of the system. For collisional systems, [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] is applied to study the microstate of a stellar system, and is also commonly used to study the different statistical ensembles of statistical mechanics.

=== Convention and notation in case of a thermal distribution ===
In most of stellar dynamics literature, it is convenient to adopt the convention that the particle mass is unity in solar mass unit <math> M_\odot</math>, hence a particle's momentum and velocity are identical, i.e.,
<math display="block"> \mathbf{p} = m \mathbf{v} = \mathbf{v}, ~ m=1, ~ N_\text{total} = M_\text{total},</math>

<math display="block"> {dM \over dx^3 dv^3} = f(\mathbf{x},\mathbf{v},t) =  f(\mathbf{x},\mathbf{p},t) \equiv {dN \over dx^3 dp^3} </math>

For example, the thermal velocity distribution of air molecules (of typically 15 times the proton mass per molecule) in a room of constant temperature <math> T_0 \sim \mathrm{300K} </math> would have a [[Maxwell distribution]]
<math display="block"> f^\text{Max}(x,y,z,m V_x,m V_y,m V_z) = {1 \over (2\pi \hbar)^3} {1 \over \exp\left({E(x,y,z,p_x,p_y,p_z) - \mu \over kT_0}\right) + 1 } </math>
<math display="block"> f^\text{Max} \sim {1 \over (2\pi \hbar/m)^3} e^{\mu \over kT_0 } e^ {-E \over m\sigma_1^2}, </math>

where the energy per unit mass <math display="block"> E/m = \Phi(x,y,z) + (V_x^2 + V_y^2 + V_z^2)/2, </math> where <math>\Phi(x,y,z) \equiv g_0 z = 0</math>

and <math display="inline"> \sigma_1 =\sqrt{k T_0/m} \sim \mathrm{0.3km/s}</math> is the width of the velocity Maxwell distribution, identical in each direction and everywhere in the room, and the normalisation constant <math> e^{\mu \over kT_0} </math> (assume the chemical potential <math display="inline">\mu \sim  (m\sigma_1^2) \ln\left[n_0 \left({\sqrt{2\pi}\hbar \over m \sigma_1}\right)^3\right] \ll 0 </math> such that the Fermi-Dirac distribution reduces to a Maxwell velocity distribution) is fixed by the constant gas number density <math>n_0 = n(x,y,0) </math> at the floor level, where 
<math display="block">
n(x,y,0) = \!\! \int_{-\infty}^\infty m dV_x \!\! \int_{-\infty}^\infty m dV_y \!\! \int_{-\infty}^\infty m dV_z f(x,y,0,mV_x,mV_y,mV_z) </math>
<math display="block"> n \approx {(2\pi)^{3/2} (m\sigma_1)^3 \over (2\pi \hbar)^3} e^{\mu \over m \sigma_1^2}.
</math>

=== 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]].

=== Probability-weighted moments and hydrostatic equilibrium ===
Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space <math display="block">
 {1 \over \rho_p } \int\! \left\{\mathbf{v}_p {d [f_p m_p]\over dt}  - \langle{\mathbf{v}}\rangle_p  {d [f_p m_p]\over dt}\right\} d^3\mathbf{v} = 0, </math> and obtain the Momentum (Jeans) Eqs. of a <math>^p</math>opulation (e.g., gas, stars, dark matter):

<math display="block">\begin{align}
\overbrace{ \left({\partial \over \partial t}+\sum_{j=1}^{3} \langle{v_j^p}\rangle {\partial  \over \partial x_j}\right) \langle{v_i^p}\rangle}^{\dot{\langle{v}\rangle}_i^p} &
\underbrace{=}_{EoM} 
\overbrace{-\partial \Phi(t,\mathbf{x})\over \partial x_i}^{g_i\sim O(-GM/R^2)} ~~
\underbrace{-}^\text{pressure}_\text{balance}~~\sum_{j=1}^{3} {\partial \over \rho^p \partial x_j}
\overbrace{[\underbrace{\rho^p(t,\mathbf{x})}_{\int_\infty\!\!\!\!m_p f_p d^3\mathbf{v}} \underbrace{\sigma_{ji}^p(t,\mathbf{x})}_{O(c_s^2)}]}^{\int\limits_\infty\!\! d\mathbf{v}^3  (\mathbf{v}_j-\langle{v}\rangle^p_j) (\mathbf{v}_i-\langle{v}\rangle^p_i)m_pf_p }
- {\underbrace{\langle{v_i^p}\rangle \overbrace{[\dot{m}_p/m_p]}^{1/t|^\text{fric}_{\text{visc}~m_p=M_\text{gas}}}}_\text{snow.plough}}, \\
0& =  -{\partial \Phi(t,\mathbf{x})\over \partial x_i} -{\partial (n \sigma^2 ) \over n \partial x_i}, ~~\text{hydrostatic isotropic velocity, no flow and friction }.\end{align}
</math>

The general version of Jeans equation, involving (3 x 3) velocity moments is cumbersome.  
It only becomes useful or solvable if we could drop some of these moments, especially drop the off-diagonal cross terms for systems of high symmetry, and also drop net rotation or net inflow speed everywhere.

The isotropic version is also called 
[[Hydrostatic equilibrium]] equation where balancing pressure gradient with gravity; the isotropic version works for axisymmetric disks as well, after replacing the derivative dr with vertical coordinate dz.  It means that we could measure the gravity (of dark matter) by observing the gradients of the velocity dispersion and the number density of stars.