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fokker-planck_equation [2013/09/17 14:20] nikolaj |
fokker-planck_equation [2013/09/17 14:32] nikolaj |
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| | @#88DDEE: $ n\in\mathbb N $ | | | @#88DDEE: $ n\in\mathbb N $ | | ||
| | @#88DDEE: $ \mu:C(\mathbb R^n,\mathbb R^n) $ | | | @#88DDEE: $ \mu:C(\mathbb R^n,\mathbb R^n) $ | | ||
| - | | @#88DDEE: $ D:C^2(\mathbb R^n,\mathbb R^{n^2}) $ | | + | | @#88DDEE: $ \mathsf{D}:C^2(\mathbb R^n,\mathbb R^{n^2}) $ | |
| + | | @#DDDDDD: $ ::\mu(\mathbf{x}) $ | | ||
| + | | @#DDDDDD: $ ::\mathsf{D}(\mathbf{x}) $ | | ||
| | @#FFBB00: $ f \in \mathrm{it} $ | | | @#FFBB00: $ f \in \mathrm{it} $ | | ||
| | @#55EE55: $ f:C^2(\mathbb R^n\times\mathbb R,\mathbb R) $ | | | @#55EE55: $ f:C^2(\mathbb R^n\times\mathbb R,\mathbb R) $ | | ||
| - | |||
| | @#DDDDDD: $ ::f(\mathbf{x},t) $ | | | @#DDDDDD: $ ::f(\mathbf{x},t) $ | | ||
| - | | @#DDDDDD: $ ::\mu(\mathbf{x}) $ | | ||
| - | | @#DDDDDD: $ ::D(\mathbf{x}) $ | | ||
| - | | @#55EE55: $ \frac{\partial }{\partial t} f = -\sum_{i=1}^n \frac{\partial}{\partial x_i} \left( \mu_i\cdot f \right) + \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial^2}{\partial x_i \, \partial x_j}\ \left( D_{ij}\cdot f \right) $ | | + | | @#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 ==== | ||
| - | We can pull out one set of derivative operators and hence rewrite the equation as | ||
| - | |||
| - | $ \frac{\partial }{\partial t} f = \sum_{i=1}^n \frac{\partial}{\partial x_i} \left( - \mu_i\cdot f + \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\ \left( D_{ij}\cdot f \right) \right) $ | ||
| - | |||
| Among other things, the Fokker-Planck equation also describes the evolution of the probability density of a [[Wiener process]]. | Among other things, the Fokker-Planck equation also describes the evolution of the probability density of a [[Wiener process]]. | ||
| ==== Parents ==== | ==== Parents ==== | ||
