If we have some potential with a mininum point, then the motion of particles will be attracted to this minimum point. With the force in mind, we can write down the master equation, which is the \(\zeta \neq 0\) case,

\[\frac{\partial}{\partial t} P(x,t) = \frac{\partial}{\partial x}\left( \frac{\partial U(x)}{\partial x} P(x,t) \right) + D \frac{\partial^2}{\partial x^2} P(x,t) .\]

This equation is called the Smoluchowski equation.

A very simple case is

\[\frac{\partial}{\partial t} P(x,t) = \gamma \frac{\partial}{\partial x}\left(x P(x,t) \right) + D \frac{\partial^2}{\partial x^2} P(x,t) .\]

which corresponds to a quadrople potential \(U(x) = \gamma x^2/2\).

Hint

We have methods of characteristics to solve such partial differential equations.

We can also use Fourier transform to solve the problem. However, we will only get

\[\frac{\partial}{\partial t} P^k = \cdots \frac{\partial}{\partial k} P^k + \cdots k^2 P^k\]

and the propagator is

\[\Pi(x,x',t) = \frac{e^{-(x - x' \exp(-\gamma t))^2}{4D\mathscr T(t)} }{\sqrt{4 \pi D \mathscr T(t)}}\]

where \(\mathscr T(t) = \frac{1-e^{-2\gamma t}}{2\gamma}\).

[1] | This is Riccati’s equation. More information here. |

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