Standard solutions of the Fokker-Planck equation#

(Based on Risken §4.4)

Continuity equation form#

Recall

Fokker-Planck in higher dimensions
\[ \frac{\partial p(t, x)}{\partial t} = \underbrace{- \sum_i \frac{\partial}{\partial x_i} \left[D_i^{(1)}(x) \, p(t,x)\right]}_{\text{drift}} + \underbrace{\sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} \left[D_{ij}^{(2)}(x) \, p(t, x) \right]}_{\text{diffusion}} \]

We can rewrite the FPE as a continuity equation

\[\begin{split}\begin{aligned} \frac{\partial p(t, x)}{\partial t} &= - \nabla \left[\left(D^{(1)}(x) + \sum_{j} \frac{\partial}{\partial x_j} D^{(2)}\right) p(t, x) \right] \\ &= - \nabla J(x) \end{aligned}\end{split}\]

where \(J(x)\) is the probability current.

Note

Since total probability is conserved, it makes sense to think of probability as flowing from one cell \([x, x + dx)\) to its neighbours.

Stationary solutions#

The continuity equation form is especially useful for finding stationary solutions to the FPE. Example stationary solutions:

  • Everything dies to a fixed point: \(p(t, x) = δ(x-x_0)\).

  • Frozen position (e.g. Galton board).

  • Circulation.

A stationary solution is defined by \(\frac{\partial p(t, x)}{\partial t} = 0\). Then

\[\begin{split}\begin{aligned} \frac{\partial p(t, x)}{\partial t} &= 0 = -\nabla J(x) \\ J(x) &= C \end{aligned}\end{split}\]

If \(x \in \mathbb{R}^n\), then the only non-trivial solution is \(J(x) = 0\). Thus we reduce the problem to a much simpler ODE:

(8)#\[ D^{(1)}(x) p(t, x) + \nabla \left(D^{(2)} p(t, x)\right) = 0 \,. \]

Going further#

Entire books are devoted to developing techniques for solving more difficult cases of the FPE; for example

  • The Fokker-Planck equation: methods of solution and applications [Risken, 1996];

  • Noise-induced transitions: theory and applications in physics, chemistry, and biology [Horsthemke and Lefever, 2006].