Fokker-Planck equation with time-dependent potential

Question

I'm reposting this from PhysicsSE since I did not receive any answers there.

Consider a Fokker-Planck (FP) equation where the advection term is a function of time, i.e. \begin{align} \frac{\partial P ( x , t )}{\partial t} = -\nabla \cdot \left[ -\mu \, P \, \nabla U (x,t) - D \nabla P \right]. \qquad\qquad ({\rm I}) \end{align} Q1 Are there general steady-state distributions (ie $\partial_t P = 0$) associated with this FP (assuming free boundary conditions)?

if in similarity with the equilibrium case, we set the probability current to zero, we obtain \begin{align} P (x,t) \propto \exp( - \mu U (x,t) / D ), \qquad\qquad ({\rm II}) \end{align} which is time-dependent, and therefore does not satisfy the FP equation.
However, I intuitively guess there could exist certain regimes that it can approximate the real solution (for example if $U$ varies slowly with time).

Q2 Under what conditions $({\rm II}$) could approximate the solution to $({\rm I})$? (note that a physical justification could also help).

Q3 In case the system (approximately) reaches the distribution given in $({\rm II})$, what sets the corresponding time-scale?‌ To clarify, I am trying to understand whether this would be a diffusive scale such as $L^2/D$ where $L$ is a typical length-scale in the system, or it would be set by the time dependence of the potential $U$.

Answer 1

I must concede that this is not a very mathematical answer.

1. You shouldn't expect any nontrivial bona fide steady states.
2. You can try to invoke a quasistatic approximation in the style that you suggested if $U$ varies slowly enough in time. But this slowness is taken relative to the mixing time for the process with $U$ frozen: you need $U$ to change very little on the time scale of mixing for the original process. That mixing can be extremely slow, especially if $\mu$ is really large compared to $D$. Also note that the variation in $U$ over time only really matters in the regions that contain significant probability. Intuitively, if after some time passes, a region of phase space increases in energy from one huge number to an even bigger number, the diffusion process doesn't notice because it essentially wasn't there to begin with.
3. The mixing time scale can be related to the relative strength of diffusion vs. the length scale of the effective "free diffusion region" (if the potential is basically constant inside some region), i.e. scaling as $L^2/D$. It can instead be related to the relative strength of diffusion vs. the strength of the trapping potential, if the diffusion is weak compared to the trapping potential. (This scenario is developed in great detail in Freidlin-Wentzell theory.) It can instead be related first to the time to wait for $U$ to make some transition, and then after that it can be related to one of the former scenarios.

I'd suggest playing around with continuous time Markov chains on finite state spaces to get some intuition, there is a lot to be learned about basically everything except stability from that setting.

Answer 2

Regarding Q1, it is hard to see how $P$ is time independent, yet $U(x,t)$ is not. Maybe for some special cases.