# Exact solution for non-linear Fokker-Planck equation with 2 variables

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I'm searching for exact (analytical) results for FP equation in 2 variables (such as $x$ and $p$ in 1D) with a steady state. Kramer's like (with force due to confining potential, such as harmonic potential) but with any nonlinear dissipation term (*or piece-wise linear*, however not $F(p) = -\gamma p$ for all $x$ and $p$). I don't need the time dependent part, only the steady state / asymptotic solution that satisfies the $L_{FP}W(x,p)=0$. For example, the following Langevin equations

$$\dot x = \omega p$$
$$\dot p = F(p)-\omega x+\sqrt{2D_p}\xi(t)$$
are equivalent to the following Fokker-Planck equation

$$[\omega(-p\partial_x+x\partial_p)+\partial_p(-F(p)+D_p\partial_p)]W(x,p)=\partial_tW(x,p)=0.$$

Notice that arbitrary choice of $F(p)$  and appropriate setting $D_p=-F(p)/p$ will satisfy the Fokker Planck equation above, as a consequence of the fluctuation-dissipation theorem. In my request I search preferably for a solution with $D=\text{const}$, thus as a consequence, inconsistent with the fluctuation dissipation - non-thermal distribution (not Boltzmann-Gibbs, also not abeying detailed balance, since fluctuation dissipation derived by assumption of detailed balance).
Any other solution which is not consistent with the fluctuation dissipation* is good as well.

* Without going into details, I've seen few papers that try to generalize the fluctuation-dissipation to non-thermal states. For my porpuses I need just inconsistensy with the "regular" one, illustated by Einstein's relation $D_p=\gamma m k_B T$ or $\frac{D_p p}{F(p)}\sim \text{const}$. Any solution that doesn't obey this relation will suffice.

EDIT:
I've discussed briefly the topic with a professor who noted that few people did some work in the past about $F(p)=-\gamma p^3$ force, and threw the name of Haye Hinrichsen. I did a brief overview of few of his papers, however didn't find anything. Maybe I missed it, and reference to a proper solution will be good as well.

Second best solution is series solution in the form of $W(x,p)=\sum_{i,j}^\infty a_{i,j} x^i p^j$ with closed formula for $a_{i,j}$ coefficients. asked Dec 11, 2018
edited Dec 15, 2018

perhaps you would be interested by Arend Scheurkogel, Isaac Elishakoff : An Exact Solution of the Fokker-Planck Equation for Nonlinear Random Vibration of a Two-Degree-of-Freedom System ( under some conditions )

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