# Exact solution for non-linear Fokker-Planck equation

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.

EDIT1: 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.

EDIT2: following the comment of @honeste_vivere I found the following article -
Dimentberg, M. F. "An exact solution to a certain non-linear random vibration problem." International Journal of Non-Linear Mechanics 17.4 (1982): 231-236. Which solves a non-linear Langevin equation -
$$\ddot{x}+\dot{x}\left(2\alpha (1+\eta(t))+\beta_1 \left(x^2+\frac{\dot{x}^2}{\Omega^2}\right)\right)+\Omega^2x(1+\xi(t))=\zeta(t)$$ with $$\eta(t)$$, $$\xi(t)$$ and $$\zeta(t)$$ as Gaussian noise, and uses Stratanovich interpretation to get to the Fokker-Planck, under certain conditions on the parameters. This solution can be seen as non-linear damping and sufficient for my purposes.

Additional solutions are still more than welcome, as they will strengthen my point.

• I don't think that this will be a suitable answer to your question, because your equation for $\dot{x}$ is not related to $F(p)$, but you might possibly find this answer or at least the references in it helpful. – user197851 Dec 11 '18 at 23:07
• Look up a paper by M.A. Malkov with doi:10.1103/PhysRevD.95.023007 published in 2017 in Phys. Rev. D. It's the only exact, analytic solution of which I am aware (but it's also only 1D). – honeste_vivere Dec 17 '18 at 0:54
• Note that you don't need to put an edit history into your question. An edit history is available for those who are interested. It would be better just to edit the question to make it one cohesive thing. Also your second edit could be sufficient enough as an answer to your own question – BioPhysicist Dec 17 '18 at 3:39
• Take a look at this book: Field theory of nonequilibrium system/chapter 4.5 (by Alex Kamenev) – Jack Dec 19 '18 at 1:19