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Neoclassical transport: Difference between revisions
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→Brief summary of the theory
m (→Limitations) |
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where <math>\alpha</math> indicates the particle species, <math>v</math> is the velocity, | where <math>\alpha</math> indicates the particle species, <math>v</math> is the velocity, | ||
<math>F</math> is a force and <math>C_\alpha</math> the Fokker-Planck collision operator. | <math>F</math> is a force (the Lorentz force acting on the particle) and <math>C_\alpha</math> the Fokker-Planck collision operator. | ||
The derivation of this collision operator is highly non-trivial and requires making specific assumptions; | The derivation of this collision operator is highly non-trivial and requires making specific assumptions; | ||
in particular it must be assumed that a single collision has a small random effect on the particle velocity, | in particular it must be assumed that a single collision has a small random effect on the particle velocity, | ||
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The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy). | The collision operator must also satisfy some obvious conservation laws (conservation of particles, momentum, and energy). | ||
Once the collision operator is decided, the moments of the Fokker-Planck equation can be computed: | Once the collision operator is decided, the moments of the Fokker-Planck equation can be computed. First, some definitions are needed: | ||
:<math> | :<math> | ||
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(energy flux) | (energy flux) | ||
:<math> | |||
P' = \int{m (v-u) \cdot (v-u) f d^3v} | |||
</math> | |||
(pressure tensor) | |||
:<math> | |||
q = \int{\frac{m (v-u)^2}{2} (v-u) f d^3v} | |||
</math> | |||
(heat flux) | |||
The main goal of Neoclassical transport theory is to provide a closed set of equations for the time evolution of these moments, for each particle species. | |||
''(Further detail needed)'' | ''(Further detail needed)'' | ||