Eigenfunctions of the Smoluchowski equation

In this Appendix we discuss the eigenfunction expansion of the Smoluchowski equation (3.21). For the sake of simplidty, we represent the set of variables jc = (xi. , Xat) by x, and write the Smoluchowski equation as 

Let Wp(x) be the right-hand eigenfunctions and il p(x) be the left-hand eigenfunctions  

It is easy to prove by direct substitution that the right-hand and left-hand eigenfunctions arc related by 

The equilibrium distribution function Ve, is an eigenfunction with eigenvalue 0, which will be denoted by the sufOx p = 0, so that V o= 1-All the other eigenvalues are positive. To show this we multiply both sides of eqn (3.1.5) by ijfpix) and integrate over jc  

Using eqn (3.1.2) and the integral by parts, the right-hand side is rewritten as 

---