# Smoluchowski Equation¶

## 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.

  This is Riccati’s equation. More information here.

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