What is the procedure to pass from the description of a phenomenon made by the Langevin equation: $$ \frac{dv}{dt}=-\frac{v}{\tau}+\sqrt{2c}\,\eta $$ to the corresponding description with the Fokker-Planck equation?
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Consider a set of equations- $$\dot{x_i}=a_i(t,\vec{x})+\xi_i(t)$$ Coordinate $x_i$ can be $x$, $p$ and such. I na vector form it can be rewritten as - $$\dot{\vec{x}}=\vec{a}(t,\vec{x})+\vec{\xi}(t)$$ Fokker Planck assumes the random force is whie Gaussian noise (there exist generalizations for other cases). Moreover,the random force in most cases considered as delta correlated in time - $$\langle \xi_i(t_1)\xi_j(t_2)\rangle=2D_i\delta_{ij}\delta(t_1-t_2)$$ The equivalent Fokker-Planck is- $$\frac{\partial \rho}{\partial t}=\underbrace{\left[-\frac{\partial}{\partial \vec{x}}\vec{a}+\frac{\partial}{\partial \vec{x}}\overleftrightarrow{D}\frac{\partial}{\partial \vec{x}}\right]}_{L_{FP}}\rho$$
Notice the reversibility of this procedure - ig you know the Fokker Planck operator you can deduce the equivalent Langevin equations.
For reference Risken's The Fokker-Planck Equation is the best but this is also quite comprehensive.
Alexander
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