Smoluchowski Equation¶

Fig. 27 Probability distribution with an attraction point.

Smoluchowski equation describes the probability distribution of particles in a attractive potential. Given a potential \(U(x)\), the master equation is,

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

This equation is called the Smoluchowski equation.

For a quadratic potential \(U(x) = \gamma x^2/2\), we get

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

Hint

The Smoluchowski equation is solved by the methods of characteristics.

Apply Fourier transform to the Smoluchowski equation, we get

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

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

Fig. 28 Examples of the normalized time parameter in the solution of Smoluchowski equation.

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