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
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