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)=−γp for all x and p). I don't need the time dependent part, only the steady state / asymptotic solution that satisfies the LFPW(x,p)=0. For example, the following Langevin equations
˙x=ωp
˙p=F(p)−ωx+√2Dpξ(t)
are equivalent to the following Fokker-Planck equation
[ω(−p∂x+x∂p)+∂p(−F(p)+Dp∂p)]W(x,p)=∂tW(x,p)=0.
Notice that arbitrary choice of F(p) and appropriate setting Dp=−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=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 Dp=γmkBT or DppF(p)∼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)=−γp3 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)=∑∞i,jai,jxipj with closed formula for ai,j coefficients.