===== Macroscopic observables from kinetic theory =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ f $ ... one-particle reduced distribution function |
| @#DDDDDD: range       | @#DDDDDD: $ N \equiv \mathrm{dim}(\mathcal M) $ |
| @#55CCEE: context     | @#55CCEE: $ q,m\in \mathbb R^*$  |

In terms of the phase space probability density, $f=f_1$ and $N$ is the number of described particles in the system with mass $m$ and charge $q$.

| @#FFBB00: definiendum | @#FFBB00: $ \langle n,\rho,u,c,\Gamma,j,J,{\mathrm p},{\mathrm P},V,C,e,E,q,Q,T \rangle \in \mathrm{it} $ | 

Number density/concentration, mass density, mean velocity, velocity deviation from the mean velocity, particle flux, current density, current, pressure, thermal velocity, energy and flux in abolute and comoving frame, and lastly temperature.

| @#AADDEE: let         | @#AADDEE: $ :: A({\bf v}) $ |
| @#AADDEE: let         | @#AADDEE: $ \langle A \rangle({\bf x},t) \equiv \int\ A({\bf v})\ f({\bf x},{\bf v},t)\ \mathrm d^3v$ | 
| @#AADDEE: let         | @#AADDEE: $ v({\bf v}):={\bf v} $ |

| @#FFFDDD: for all     | @#FFFDDD: $i,j\in\{1,2,3\}$ |

| @#FFBB00: definiendum | @#FFBB00: $ n := \langle \mathrm{1} \rangle $ |
| @#FFBB00: definiendum | @#FFBB00: $ \rho := m\ n $ |

| @#FFBB00: definiendum | @#FFBB00: $ u := \langle v \rangle $ |
| @#FFBB00: definiendum | @#FFBB00: $ c({\bf x},{\bf v},t) := {\bf v}-u $ | 
| @#FFBB00: definiendum | @#FFBB00: $ \Gamma := n\ u $ | 
| @#FFBB00: definiendum | @#FFBB00: $ j := q\ u $ | 
| @#FFBB00: definiendum | @#FFBB00: $ J := n\ j $ | 

| @#FFBB00: definiendum | @#FFBB00: $ {\mathrm p}_{ij} := \rho\ \langle v_i\ v_j \rangle $ |
| @#FFBB00: definiendum | @#FFBB00: $ {\mathrm P}_{ij} := \rho\ \langle c_i\ c_j \rangle $ |  
| @#FFBB00: definiendum | @#FFBB00: $ V := \langle v^2 \rangle^{\frac{1}{2}} $ |
| @#FFBB00: definiendum | @#FFBB00: $ C := \langle c^2 \rangle^{\frac{1}{2}} $ |
| @#FFBB00: definiendum | @#FFBB00: $ e := \rho \frac{1}{2}V^2 $ | 
| @#FFBB00: definiendum | @#FFBB00: $ E := \rho \frac{1}{2}C^2 $ | 

| @#FFBB00: definiendum | @#FFBB00: $ q_i := \rho \frac{1}{2}\langle v^2\ v_i\rangle $ | 
| @#FFBB00: definiendum | @#FFBB00: $ Q_i := \rho \frac{1}{2}\langle c^2\ c_i\rangle $ | 
| @#FFBB00: definiendum | @#FFBB00: $ T := E/\left(\frac{3}{2}n\ k_B\right) $ | 

==== Discussion ====
The pressure tensor is (proportional to) the covariance matrix of $f$ w.r.t $v$ and the energy is the variance.

== Theorems ==
^ $\langle c\rangle = 0$ ^
^ $J=q\ \Gamma$ ^
^ $E=\tfrac{1}{2}\mathrm{tr}(\mathrm{P})$ ^

The frame with velocity $c$ is moving with the fluid. The advantage of this frame is that $\langle c\rangle$, the disadvantage is that $c$ (in contrast to any $v$) is a function of ${\bf x}$.

The above observables inherit time evolution equations from the Boltzmann equation. In the frame with $\langle c\rangle$ they read

^ $ \rho \left( \frac{\partial}{\partial t} + u\ \mathrm{div} \right) u = - \mathrm{div}(P) \rho K $ ^
^ $ \frac{\partial}{\partial t} E + \nabla (u\ E) = - \mathrm{tr}(P\cdot \mathrm{grad}(u)) - \mathrm{div}(Q) $ ^

In a general frame, they look a little shorter, because some partial derivatives vanish.

After plugging in some transport coefficients relating $P$ in terms of $u$, one obtains the [[Navier-Stokes equations]].

== Boltzmann equation ==
A notable solution of an instance of the Boltzmann equation is the //local Maxwellian//:

^ $ f_0({\bf x},{\bf v},t) = \frac{n({\bf x},t)}{\left( 2\pi\ k_B T({\bf x},t) \right)^{\frac{3}{2}}} \text{exp}\left( -m\frac{1}{2}c({\bf x},{\bf v},t)^2/(k_B T({\bf x},t)) \right) $ ^

Another one is the //absolute Druyvesteyn distribution// for electrons in a mass $M$ ion background and with a constant external electrical field $K$:

| @#DDDDDD: range       | @#DDDDDD: $ s \equiv (M/6m)\left(\frac{K\ l}{k_B T}\right)^2 $ | 

The mean free path $l$ appears as a parameter.

^ $ f_D({\bf x},{\bf v}) \propto \left(m\frac{1}{2}{\bf v}^2/(k_B T)+s\right)^s\mathrm{exp}\left(-m\frac{1}{2}{\bf v}^2/(k_B T)\right) $ ^

In the appropriate scenarios, both of the above distribution reduce to the Maxwell-Boltzmann distribution

^ $ f_0({\bf v}) = \frac{n}{\left( 2\pi\ k_B T \right)^{\frac{3}{2}}} \text{exp}\left( -m\frac{1}{2}c({\bf v})^2/(k_B T) \right) $ ^

=== Reference ===
Wikipedia: 
[[https://en.wikipedia.org/wiki/Boltzmann_equation|Boltzmann equation]]

==== Parents ====
=== Context ===
[[Reduced distribution function]]
=== Related ===
[[Boltzmann equation]]