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?

  • $\begingroup$ What beyond the linked Wikipedia article is this question about? A substitution gives you $\partial_t p = \frac{1}{\tau}\partial_v(vp) + c\partial_v^2p$ $\endgroup$ – alarge Feb 27 '18 at 22:53
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    $\begingroup$ It's the substitution itself and, moreover, how to pass from a stochastic description of the $x$ and $v$ (Langevin) to a deterministic description of the $P(x)$ and $P(v)$ (Fokker-Planck) $\endgroup$ – Drebin J. Mar 1 '18 at 10:33
  • $\begingroup$ The substitution of your equation is just a direct drop into the formula in Wikipedia, as in my previous comment. If you want $p(x)$, you'll e.g. just write the system of equations for $dx$ and $dv$, get $p(x,v)$ as a substitution to the formula on Wikipedia and integrate over $v$. As for how to derive Fokker-Planck, that's in any standard textbook and not really a conceptual question fitting this website. $\endgroup$ – alarge Mar 1 '18 at 11:47

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.

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