Problem set 2:

Consider a chemical reaction in which a certain species X is produced at a steady rate $\beta\Omega$. If two molecules collide they produce a new inert substance which is removed from the system $(2X\rightarrow E)$. Let the collision rate be $\alpha n(n-1)/\Omega$. The reactions are irreversible and we write the master equation on the form

$$\frac{\partial P(n,t)}{\partial t}=\Omega\beta(E^{-1}-1)P(n,t) +\frac{\alpha}{\Omega}(E^2-1)n(n-1)P(n,t)$$

Where E is the raising (or creation) operator.

• Carry out a system size expansion of the master equation to derive a Fokker-Planck equation for $\Pi(x,t)$ where
  
  $$n=\phi\Omega +x\sqrt{\Omega}$$
  
  

• Find the steady concentration $\phi$ and the variance of x to leading order in $\Omega$.
• The problem is actually exactly solvable using the method of characteristics discussed in Lecture 6. The solution is outlined in a paper by Mazo [1975]. Fill in the steps of Mazo's calculation and show that the Fokker-Planck equation and the exact solution agree to leading order. In particular justify the boundary conditions for the generating function!
• Find the correction of order unity to the mean number of X molecules.

Return to table of content