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Neoclassical transport: Difference between revisions
→Brief summary of the theory
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== Brief summary of the theory == | == Brief summary of the theory == | ||
The theory starts from the Fokker-Planck Equation for the particle distribution function <math> | The theory starts from the (Markovian) Fokker-Planck Equation for the particle distribution function <math>f_\alpha(x,v,t)</math>: | ||
<math> | <math> | ||
\frac{\partial | \frac{\partial f_\alpha}{\partial t} + v\cdot \nabla f_\alpha + F \frac{\partial f_\alpha}{\partial v} = C_\alpha(f) | ||
</math> | </math> | ||
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. | |||
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, | |||
and that the collisions are sufficiently frequent for the resulting particle trajectory to be described as a random walk. | |||
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: | |||
<math> | |||
n u = \int{v f d^3v} | |||
</math> | |||
(particle flux) | |||
<math> | |||
P = \int{m v \cdot v f d^3v} | |||
</math> | |||
(stress tensor) | |||
<math> | |||
Q = \int{\frac{m v^2}{2} v f d^3v} | |||
</math> | |||
(energy flux) | |||
''(Further detail needed)'' | ''(Further detail needed)'' | ||