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boltzmann_equation [2014/06/26 14:46] nikolaj |
boltzmann_equation [2015/06/15 17:48] nikolaj |
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| @#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]$ | | | @#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 $\dots = \sum_{j=1}^S J[f_i|f_j]$ here --- check that. | + | >I'm not sure about the summation "$\cdots = \sum_{j=1}^S J[f_i|f_j]$" here --- check that. |
==== Discussion ==== | ==== Discussion ==== | ||
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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: |