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boltzmann_equation [2013/12/17 23:15] nikolaj |
boltzmann_equation [2015/06/15 17:48] nikolaj |
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| ===== Boltzmann equation ===== | ===== Boltzmann equation ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ {\bf K}:\mathbb R^3\times\mathbb R^3\times\mathbb t\to\mathbb R $ | | + | | @#55CCEE: context | @#55CCEE: $ {\bf K}:\mathbb R^3\times\mathbb R^3\times\mathbb t\to\mathbb R $ | |
| - | | @#DDDDDD: $ :: {\bf K}({\bf x},{\bf v},t)$ | | + | | @#DDDDDD: range | @#DDDDDD: $ :: {\bf K}({\bf x},{\bf v},t)$ | |
| - | | @#88DDEE: $ S\in\mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $ S\in\mathbb N $ | |
| | $i,j\in\text{range}(S)$ | | | $i,j\in\text{range}(S)$ | | ||
| - | | @#88DDEE: $ m_i\in \mathbb R^* $ | | + | | @#55CCEE: context | @#55CCEE: $ m_i\in \mathbb R^* $ | |
| - | | @#88DDEE: $ I_{ij}: \mathbb R^3\times[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\times[0,2\pi]\to\mathbb R $ | | + | | @#55CCEE: context | @#55CCEE: $ I_{ij}: \mathbb R^3\times[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\times[0,2\pi]\to\mathbb R $ | |
| - | | @#DDDDDD: $ :: I_{ij}({\bf v},\vartheta,\varphi) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ :: I_{ij}({\bf v},\vartheta,\varphi) $ | |
| The integer $S$ denote the species, $m_i$ their respective masses and $I_{ij}$ the differential cross sections for two particle collisions. | The integer $S$ denote the species, $m_i$ their respective masses and $I_{ij}$ the differential cross sections for two particle collisions. | ||
| - | | @#88DDEE: $ {\bf v'},\ {\bf v'}_1 : \mathbb R^{2\times 3}\to \mathbb R^3 $ | | + | | @#55CCEE: context | @#55CCEE: $ {\bf v'},\ {\bf v'}_1 : \mathbb R^{2\times 3}\to \mathbb R^3 $ | |
| - | | @#DDDDDD: $ :: {\bf v'}({\bf v'},{\bf v}_1),\ {\bf V}_1({\bf v},{\bf v}_1) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ :: {\bf v'}({\bf v'},{\bf v}_1),\ {\bf V}_1({\bf v},{\bf v}_1) $ | |
| - | | @#FFBB00: $ {\bf f} \in \mathrm{it} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ {\bf f} \in \mathrm{it} $ | |
| - | | @#55EE55: $ f_i:\mathbb R^3\times\mathbb R^3\times\mathbb R\to\mathbb R_+ $ | | + | | @#55EE55: postulate | @#55EE55: $ f_i:\mathbb R^3\times\mathbb R^3\times\mathbb R\to\mathbb R_+ $ | |
| - | | @#DDDDDD: $ :: f_i({\bf x},{\bf v},t)$ | | + | | @#DDDDDD: range | @#DDDDDD: $ :: f_i({\bf x},{\bf v},t)$ | |
| - | | @#DDDDDD: $ J[f_i|f_j]({\bf x},{\bf v},t) \equiv \int\int g\ I_{ij}(g,\vartheta,\varphi)\ \left(f_i({\bf x},{\bf v'}({\bf v},{\bf v}_1),t)\cdot f_j({\bf x},{\bf v'}_1({\bf v},{\bf v}_1),t)-f_i({\bf x},{\bf v},t)\cdot f_j({\bf x},{\bf v}_1,t)\right)\ \mathrm d\Omega(\vartheta,\varphi)\ \mathrm d^3v_1 $ | | + | | @#DDDDDD: range | @#DDDDDD: $ J[f_i|f_j]({\bf x},{\bf v},t) \equiv \int\int g\ I_{ij}(g,\vartheta,\varphi)\ \left(f_i({\bf x},{\bf v'}({\bf v},{\bf v}_1),t)\cdot f_j({\bf x},{\bf v'}_1({\bf v},{\bf v}_1),t)-f_i({\bf x},{\bf v},t)\cdot f_j({\bf x},{\bf v}_1,t)\right)\ \mathrm d\Omega(\vartheta,\varphi)\ \mathrm d^3v_1 $ | |
| - | | @#55EE55: $ \left(\frac{\mathrm \partial}{\partial t}+{\bf v}\cdot\nabla_{\bf x}+\frac{1}{m_i}{\bf K}\cdot\nabla_{\bf v}\right)f_i = \sum_{k=1}^S J[f_i|f_j]$ | | + | | @#55EE55: postulate | @#55EE55: $ \left(\frac{\mathrm \partial}{\partial t}+{\bf v}\cdot\nabla_{\bf x}+\frac{1}{m_i}{\bf K}\cdot\nabla_{\bf v}\right)f_i = \sum_{j=1}^S J[f_i|f_j]$ | |
| + | |||
| + | >I'm not sure about the summation "$\cdots = \sum_{j=1}^S J[f_i|f_j]$" here --- check that. | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
| Line 42: | Line 44: | ||
| at the fixed point velocity ${\bf v}$. The second term is more involved, since it doesn't represent the loss at ${\bf v}$, but the gain: It's the sum of processes of pairs particles with velocities ${\bf v'},{\bf v'}_1$, which end up with particles having the velocity ${\bf v}$ and any other velocity ${\bf v}_1$. | at the fixed point velocity ${\bf v}$. The second term is more involved, since it doesn't represent the loss at ${\bf v}$, but the gain: It's the sum of processes of pairs particles with velocities ${\bf v'},{\bf v'}_1$, which end up with particles having the velocity ${\bf v}$ and any other velocity ${\bf v}_1$. | ||
| + | === Note === | ||
| + | <code> | ||
| + | $Assumptions = {kT > 0, n > -1, \[CapitalTheta] > 0}; | ||
| + | |||
| + | f[EE_] = E^(-(EE/kT))* | ||
| + | E^(-(EE^2/\[CapitalTheta]^2))/(E^(\[CapitalTheta]^2/(2 kT)^2) | ||
| + | Sqrt[\[Pi]] /2 \[CapitalTheta] Erfc[\[CapitalTheta]/(2 kT)]); | ||
| + | |||
| + | Integrate[f[EE] EE^n, {EE, 0, \[Infinity]}] | ||
| + | </code> | ||
| === Reference === | === Reference === | ||
| Wikipedia: | Wikipedia: | ||
| Line 49: | Line 61: | ||
| === Subset of === | === Subset of === | ||
| [[PDE system]] | [[PDE system]] | ||
| - | === Context === | + | === Related === |
| [[BBGKY hierarchy]] | [[BBGKY hierarchy]] | ||
