Kinetic Theory – Fokker-Planck Equation

Abstract

In this chapter we consider a model system (protein) interacting with a surrounding medium which is only taken implicitly into account. We are interested in the dynamics on a time scale slower than the medium fluctuations. The interaction with the medium is described approximately as the sum of an average force and a stochastic force. We discuss the stochastic differential equation for 1-dimensional Brownian motion and derive the corresponding Fokker-Planck equation. We consider motion of a particle under the influence of an external force and derive the Klein-Kramers equation for diffusion in an external potential and the Smoluchowski equation as its large-friction limit. Finally we discuss the connection to the general Master equation for the probability density.

Problems

7.1

Smoluchowski Equation

Consider a 1-dimensional random walk. At times $t_{n}=n\varDelta t$ a particle at position $x_{j}=j\varDelta x$ jumps either to the left side $j-1$ with probability $w_{j}^{-}$ or to the right side $j+1$ with probability $w_{j}^{+}=1-w_{j}^{-}$. The probability to find a particle at site j at the time $n+1$ is then given by

$$ P_{n+1,j}=w_{j-1}^{+}P_{n, j-1}+w_{j+1}^{-}P_{n, j+1}. $$

Show that in the limit of small $\varDelta x,\varDelta t$ the probability distribution P(t, x) obeys a Smoluchowski equation

7.2

Eigenvalue Solution to the Smoluchowski Equation

Consider the 1-dimensional Smoluchowski equation

$$ \frac{\partial W(x, t)}{\partial t}=\frac{1}{m\gamma }\frac{\partial }{\partial x}\left[ k_{B}T\frac{\partial }{\partial x}+\frac{\partial U}{\partial x}\right] W(x, t)=-\frac{\partial }{\partial x}S(x) $$

for a harmonic potential

$$ U(x)=\frac{m\omega ^{2}}{2}x^{2}. $$

Show that the probability current can be written as

$$ S(x,t)=-\frac{k_{B}T}{m\gamma }\mathrm{e}^{-U(x)/kT}\left( \frac{\partial }{\partial x}\mathrm{e}^{U(x)/k_{B}T}W(x, t)\right) $$

and that the Fokker–Planck operator can be written as

$$ \mathfrak {L}_{FP}=\frac{1}{m\gamma }\frac{\partial }{\partial x}\left[ k_{B}T\frac{\partial }{\partial x}+\frac{\partial U}{\partial x}\right] =\frac{k_{B}T}{m\gamma }\frac{\partial }{\partial x}\mathrm{e}^{-U(x)/k_{B}T}\frac{\partial }{\partial x}\mathrm{e}^{U(x)/k_{B}T} $$

and can be transformed into a hermitian operator by

$$ \mathfrak {L}=\mathrm{e}^{U(x)/2k_{B}T}\mathfrak {L}_{FP}\mathrm{e}^{-U(x)/2k_{B}T}. $$

Solve the eigenvalue problem

$$ \mathfrak {L}\psi _{n}(x)=\lambda _{n}\psi _{n}(x) $$

and use the function $\psi _{0}(x)$ to construct a special solution

$$ W(x, t)=\mathrm{e}^{\lambda _{0}t}\mathrm{e}^{-U(x)/2k_{B}T}\psi _{0}(x). $$

7.3

Diffusion Through a Membrane

A membrane with M pore-proteins separates two half-spaces A and B. An ion X may diffuse through M pore proteins in the membrane from A to B or vice versa. The rate constants for the formation of the ion-pore complex are $k_{A}$ and $k_{B}$ respectively, while $k_{m}$ is the constant for the decay of the ion-pore complex independent of the side to which the ion escapes. Let $P_{N}(t)$ denote the probability that there are N ion-pore complexes at time t. The master equation for the probability is

$$ \frac{\mathrm{d}P_{N}(t)}{\mathrm{d}t}=-\left[ (k_{A}+k_{B})(M-N)+2k_{m}N\right] P_{N}(t)\,+ $$

$$ +\,(k_{A}+k_{B})(M-N+1)P_{N-1}(t)+2k_{m}(N+1)P_{N+1}(t). $$

Calculate mean and variance of the number of complexes

$$ \overline{N}=\sum _{N}NP_{N} $$

$$ \sigma ^{2}=\overline{N^{2}}-\overline{N}^{2} $$

and the mean diffusion current

$$ J_{AB}=\frac{\mathrm{d}N_{B}}{\mathrm{d}t}-\frac{\mathrm{d}N_{A}}{\mathrm{d}t} $$

where $N_{A, B}$ is the number of ions in the upper or lower half-space.