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macroscopic_observables_from_kinetic_theory [2014/02/22 17:12] nikolaj |
macroscopic_observables_from_kinetic_theory [2014/03/21 11:11] (current) |
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| ===== Macroscopic observables from kinetic theory ===== | ===== Macroscopic observables from kinetic theory ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $ f $ ... one-particle reduced distribution function | | + | | @#55CCEE: context | @#55CCEE: $ f $ ... one-particle reduced distribution function | |
| - | | @#DDDDDD: $ N \equiv \mathrm{dim}(\mathcal M) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ N \equiv \mathrm{dim}(\mathcal M) $ | |
| - | | @#88DDEE: $ q,m\in \mathbb R^*$ | | + | | @#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 $q$. | 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: $ \langle n,\rho,u,c,\Gamma,j,J,{\mathrm p},{\mathrm P},V,C,e,E,q,Q,T \rangle \in \mathrm{it} $ | | + | | @#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. | 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: $ :: A({\bf v}) $ | | + | | @#AADDEE: let | @#AADDEE: $ :: A({\bf v}) $ | |
| - | | @#AADDEE: $ \langle A \rangle({\bf x},t) \equiv \int\ A({\bf v})\ f({\bf x},{\bf v},t)\ \mathrm d^3v$ | | + | | @#AADDEE: let | @#AADDEE: $ \langle A \rangle({\bf x},t) \equiv \int\ A({\bf v})\ f({\bf x},{\bf v},t)\ \mathrm d^3v$ | |
| - | | @#AADDEE: $ v({\bf v}):={\bf v} $ | | + | | @#AADDEE: let | @#AADDEE: $ v({\bf v}):={\bf v} $ | |
| - | | @#FFBB00: $ n := \langle \mathrm{1} \rangle $ | | + | | @#FFFDDD: for all | @#FFFDDD: $i,j\in\{1,2,3\}$ | |
| - | | @#FFBB00: $ \rho := m\ n $ | | + | |
| - | | @#FFBB00: $ u := \langle v \rangle $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ n := \langle \mathrm{1} \rangle $ | |
| - | | @#FFBB00: $ c({\bf x},{\bf v},t) := {\bf v}-u $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \rho := m\ n $ | |
| - | | @#FFBB00: $ \Gamma := n\ u $ | | + | |
| - | | @#FFBB00: $ j := q\ u $ | | + | |
| - | | @#FFBB00: $ J := n\ j $ | | + | |
| - | | @#FFBB00: $ {\mathrm p}_{ij} := \rho\ \langle v_i\ v_j \rangle $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ u := \langle v \rangle $ | |
| - | | @#FFBB00: $ {\mathrm P}_{ij} := \rho\ \langle c_i\ c_j \rangle $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ c({\bf x},{\bf v},t) := {\bf v}-u $ | |
| - | | @#FFBB00: $ V := \langle v^2 \rangle^{\frac{1}{2}} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \Gamma := n\ u $ | |
| - | | @#FFBB00: $ C := \langle c^2 \rangle^{\frac{1}{2}} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ j := q\ u $ | |
| - | | @#FFBB00: $ e := \rho \frac{1}{2}V^2 $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ J := n\ j $ | |
| - | | @#FFBB00: $ E := \rho \frac{1}{2}C^2 $ | | + | |
| - | | @#FFBB00: $ T := E/\left(\frac{3}{2}n\ k_B\right) $ | | + | | @#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 $ | | ||
| - | | @#FFFDDD: $i,j\in\{1,2,3\}$ | | + | | @#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: $ q_i := \rho \frac{1}{2}\langle v^2\ v_i\rangle $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ T := E/\left(\frac{3}{2}n\ k_B\right) $ | |
| - | | @#FFBB00: $ Q_i := \rho \frac{1}{2}\langle c^2\ c_i\rangle $ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
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| In a general frame, they look a little shorter, because some partial derivatives vanish. | 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]]. | + | After plugging in some transport coefficients relating $P$ in terms of $u$, one obtains the [[Navier-Stokes equations]]. |
| - | + | ||
| - | >[[Navier–Stokes equations]] ... Y U no link?! | + | |
| == Boltzmann equation == | == Boltzmann equation == | ||
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| Another one is the //absolute Druyvesteyn distribution// for electrons in a mass $M$ ion background and with a constant external electrical field $K$: | Another one is the //absolute Druyvesteyn distribution// for electrons in a mass $M$ ion background and with a constant external electrical field $K$: | ||
| - | | @#DDDDDD: $ s \equiv (M/6m)\left(\frac{K\ l}{k_B T}\right)^2 $ | | + | | @#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. | The mean free path $l$ appears as a parameter. | ||
