Macroscopic observables from kinetic theory

Set

context	$f$ … one-particle reduced distribution function
range	$N \equiv \mathrm{dim}(\mathcal M)$
context	$q,m\in \mathbb R^*$

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 $q$ .

definiendum

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

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.

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

for all

$i,j\in\{1,2,3\}$

definiendum	$n := \langle \mathrm{1} \rangle$
definiendum	$\rho := m\ n$

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

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

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

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 $K$ :

range

$s \equiv (M/6m)\left(\frac{K\ l}{k_B T}\right)^2$

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: Boltzmann equation

Parents

Context

Reduced distribution function

Boltzmann equation