I have a master equation -
$$\frac{\partial P(n,t)}{\partial t}=\frac{1}{2}[(n+1)Pr(n+1,t)+(N-n+1)Pr(n-1,t)-NPr(n,t)]$$
I would like to take this master equation and convert it into a Fokker-Planck equation. I consider a variable $x=\frac{2n}{N}$ so that at steady state I would get $<x>=1$.
I tried doing some manipulations on this equation to get to the form where I can perform a taylor expansion, but I'm just not getting there. Any tips would be greatly appreciated.
I consider $N = N-n+n+1-1 $ to simplify some terms..
$$\frac{\partial P(n,t)}{\partial t}=\frac{1}{2}[(n+1)Pr(n+1,t)+(N-n+1)Pr(n-1,t)-(N-)Pr(n,t)]$$
$$\frac{\partial P(n,t)}{\partial t}=\frac{1}{2}[(n+1)Pr(n+1,t)-(n+1)Pr(n,t)]+\frac{1}{2}[(N-n+1)Pr(n-1,t)-(N-n-1)Pr(n,t)]$$
^ Can't simplify from here...
Different try?
$$\frac{\partial P(n,t)}{\partial t}=\frac{1}{2}[(n+1)Pr(n+1,t)-nPr(n,t)]+\frac{1}{2}[(N-n+1)Pr(n-1,t)-(N-n)Pr(n,t)]$$
Not sure how to take care of this either.