# Smoluchowski Equation¶

## Smoluchowski Equation¶

Fig. 11 Probability distribution with an attraction point.

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

Fig. 12 The redefined time parameter in the solution of Smoluchowski equation example.