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?

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 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 $U$.


Q1 As you have shown yourself, this equation does not have a steady-state distribution: if we set $\partial_t P = 0$, i.e., if we assume that the solution is time-independent, we still obtain a solution that depends on time, contradicting our assumption.

Q2 and Q3 In some situations one could indeed approximate the solution using form (II). The conditions can be obtained by substituting this form to the original equation and demanding that the residual term is small. One could consider, for example, a situation of approaching this quasi-equilibrium by using conjecture $$ P(x,t) = C\exp\left[-\mu U(x,t)/D\right] + p(x,t), $$ where $p(x,t)$ describes the deviation from the quasi-equilibrium. One could then study whether this perturbation have enough time to dissipate on the time scale of potential $U(x,t)$, which will depend on the diffusion coefficient, the spatial scale of $U(x,t)$, and other spatial scales in the system (e.g., those set by the boundary conditions).

The separation of times is a frequently encountered concept in relation to FPE, notably in the context of diffusion escape from a potential minimum, where the two main scales are the fast establishment of a quasi-equilibroum near the potential minimum, and the slow escape from this minimum. Notably, this problem was extensively studied with inclusion of time-dependent potential barrier height, making it a particular case of the problem discussed here - you may look up Dykman's publications in Physical Reviews, and follow up the references and the citing paper.

  • $\begingroup$ Thanks for your answer. Regarding Q1, I don't fully agree; what I showed is that there is no solution for which the probability current is zero. There may still exist time-independent solutions with nonzero currents for which the divergence of the current is zero. $\endgroup$ – SaMaSo Oct 10 '20 at 15:13
  • $\begingroup$ I am not sure that I understand you: nonzero current means that the solution is time dependent and vice versa. What probably remained unmentioned in my answer is the normalization - there are some scenarios, where it could play a role. $\endgroup$ – Vadim Oct 10 '20 at 15:25
  • $\begingroup$ Thus, one could think about some special examples with absorbing boundary conditions, or sources and sinks... $\endgroup$ – Vadim Oct 10 '20 at 15:39
  • $\begingroup$ Well, in principle you could have $\nabla\cdot\mathbf{J} = 0$ without $\mathbf{J}=0$. This could correspond to a steady-state solution (rather than an equilibrium solution where $\mathbf{J}$ is identical to zero). $\endgroup$ – SaMaSo Oct 10 '20 at 15:44
  • $\begingroup$ in case it was not already clear, $\mathbf{J} = - \mu P \nabla U -D \nabla P$ is what I call probability current. $\endgroup$ – SaMaSo Oct 10 '20 at 15:46

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