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macroscopic_observables_from_kinetic_theory [2014/02/13 16:13] 127.0.0.1 external edit |
macroscopic_observables_from_kinetic_theory [2014/02/22 17:14] nikolaj |
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| 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. | ||
| - | | @#DDDDDD: $ :: A({\bf v}) $ | | + | | @#AADDEE: $ :: A({\bf v}) $ | |
| - | | @#DDDDDD: $ \langle A \rangle({\bf x},t) \equiv \int\ A({\bf v})\ f({\bf x},{\bf v},t)\ \mathrm d^3v$ | | + | | @#AADDEE: $ \langle A \rangle({\bf x},t) \equiv \int\ A({\bf v})\ f({\bf x},{\bf v},t)\ \mathrm d^3v$ | |
| - | | $i,j\in\{1,2,3\}$ | | + | | @#AADDEE: $ v({\bf v}):={\bf v} $ | |
| - | | @#DDDDDD: $ v({\bf v}):={\bf v} $ | | + | |
| + | | @#FFFDDD: $i,j\in\{1,2,3\}$ | | ||
| | @#FFBB00: $ n := \langle \mathrm{1} \rangle $ | | | @#FFBB00: $ n := \langle \mathrm{1} \rangle $ | | ||
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| | @#FFBB00: $ q_i := \rho \frac{1}{2}\langle v^2\ v_i\rangle $ | | | @#FFBB00: $ q_i := \rho \frac{1}{2}\langle v^2\ v_i\rangle $ | | ||
| | @#FFBB00: $ Q_i := \rho \frac{1}{2}\langle c^2\ c_i\rangle $ | | | @#FFBB00: $ Q_i := \rho \frac{1}{2}\langle c^2\ c_i\rangle $ | | ||
| - | |||
| | @#FFBB00: $ T := E/\left(\frac{3}{2}n\ k_B\right) $ | | | @#FFBB00: $ T := E/\left(\frac{3}{2}n\ k_B\right) $ | | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | The pressure tensor is (propotional to) the covarance matrix of $f$ w.r.t $v$ and the energy is the variance. | + | The pressure tensor is (proportional to) the covariance matrix of $f$ w.r.t $v$ and the energy is the variance. |
| == Theorems == | == Theorems == | ||
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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 == | ||
