Schrodinger equation in term of Fokker-Planck equation

Question

From Wikipedia on the Fokker-Planck equation:

$$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\partial x}+D_2 \left( x,t\right) \frac{\partial^2}{\partial x^2}\right] \delta\left( x^{\prime}-x\right) \right) f\left( x,t\right).\qquad $$

Integrate over a time interval $\varepsilon$,

$$f\left( x^\prime ,t+\varepsilon \right) $$ $$~=~\int_{-\infty }^\infty \, dx\left(\left( 1+\varepsilon \left[ D_{1}\left(x,t\right) \frac{\partial }{\partial x}+D_{2}\left( x,t\right) \frac{\partial^{2}}{\partial x^{2}}\right]\right) \delta \left( x^\prime - x\right) \right) f\left( x,t\right)$$ $$\tag{2}+O\left( \varepsilon ^{2}\right).\qquad $$

OK, but Fokker-Planck equation for one dimension is usually

$$\tag{0} \frac{\partial}{\partial t}f(x,t) = -\frac{\partial}{\partial x}\left[\mu(x,t)f(x,t)\right] + \frac{\partial^2}{\partial x^2}\left[ D(x,t)f(x,t)\right].$$

I was not able to understand how one gets from the original equation (0) to the above (1) and how does the first equation (1) lead to the second equation (2). Can anyone explain this?

Answer 1

Hints:

$\underline{(0) \Rightarrow (1)}$: Don't try to accomplish everything at once. Do it slowly in as many steps as you need to be sure that you are calculating correctly and understand everything. The trick is to integrate by part. Be very careful to keep track of what depends on $x$ and what depends on $x^{\prime}$.

$\underline{(1) \Rightarrow (2)}$: Use Taylor series

$$f( x^{\prime} ,t+\varepsilon ) ~=~f( x^{\prime} ,t) +\varepsilon\frac{\partial }{\partial t}f\left( x^{\prime },t\right) +{\cal O}\left( \varepsilon ^{2}\right).$$