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fokker-planck_equation [2013/09/17 14:30] nikolaj |
fokker-planck_equation [2014/03/21 11:11] (current) |
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===== Fokker-Planck equation ===== | ===== Fokker-Planck equation ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $ n\in\mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $ n\in\mathbb N $ | |
- | | @#88DDEE: $ \mu:C(\mathbb R^n,\mathbb R^n) $ | | + | | @#55CCEE: context | @#55CCEE: $ \mu:C(\mathbb R^n,\mathbb R^n) $ | |
- | | @#88DDEE: $ D:C^2(\mathbb R^n,\mathbb R^{n^2}) $ | | + | | @#55CCEE: context | @#55CCEE: $ \mathsf{D}:C^2(\mathbb R^n,\mathbb R^{n^2}) $ | |
+ | | @#DDDDDD: range | @#DDDDDD: $ ::\mu(\mathbf{x}) $ | | ||
+ | | @#DDDDDD: range | @#DDDDDD: $ ::\mathsf{D}(\mathbf{x}) $ | | ||
- | | @#FFBB00: $ f \in \mathrm{it} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ f \in \mathrm{it} $ | |
- | | @#55EE55: $ f:C^2(\mathbb R^n\times\mathbb R,\mathbb R) $ | | + | | @#55EE55: postulate | @#55EE55: $ f:C^2(\mathbb R^n\times\mathbb R,\mathbb R) $ | |
+ | | @#DDDDDD: range | @#DDDDDD: $ ::f(\mathbf{x},t) $ | | ||
- | | @#DDDDDD: $ ::f(\mathbf{x},t) $ | | + | | @#55EE55: postulate | @#55EE55: $ \frac{\partial }{\partial t} f = -\mathrm{div} (\mu \cdot f) + \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial^2}{\partial x_i \, \partial x_j} (\mathsf{D}_{ij}\cdot f) $ | |
- | | @#DDDDDD: $ ::\mu(\mathbf{x}) $ | | + | |
- | | @#DDDDDD: $ ::\mathsf{D}(\mathbf{x}) $ | | + | |
- | + | ||
- | | @#55EE55: $ \frac{\partial }{\partial t} f = -\mathrm{div} (\mu \cdot f) + \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial^2}{\partial x_i \, \partial x_j}\ (\mathsf{D}_{ij}\cdot f) $ | | + | |
==== Discussion ==== | ==== Discussion ==== |