"Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"

Question

I follow the notation used in "stochastic processes in physics and chemistry"(p.245) by Van Kampen.
Left hand side of master equation is $$\frac{\partial P(n,t)}{\partial t}=\cdots.$$ We split $n$ into 2 elemnts $$n = \Omega \phi(t) + \Omega ^{1/2}\xi$$ and rewrite the first to $$\frac{\partial P(n,t)}{\partial t}=\frac{\partial}{\partial t}\Pi(\xi,t)\left|\frac{\partial \xi}{\partial n}\right|=\Omega ^{-1/2}[ \frac{\partial \Pi}{\partial t}+\frac{\partial \xi}{\partial t} \frac{\partial \Pi}{\partial \xi}]= \Omega ^{-1/2}[ \frac{\partial \Pi}{\partial t}-\Omega^{1/2} \frac{d\phi}{dt} \frac{\partial \Pi}{\partial \xi}].$$ Here my question. Why is n held constant? I think n is time dependent and should be defferenciated. I have refered to several texts but they don't mention this much. In Kampen's text there is a line like

"The time derivative in (1.4) is taken with constant n; that means in the $ξ, t$ plane along the direction given by $dξ/dt = ーΩ^(1/2) dφ/dt$."

But I cannot see what he is meaning.
Can anyone help me? Give me any physical meaning of this.

Answer 1

When you have a master equation for $P(n,t)$ is describes how the probability of a certain state (in this case the state corresponds to the particle number n) changes over time. It is important to note that the states are $\textbf{fixed}$ and the probability changes. Therefore, you expect $n$ to be a constant in time. This should not change even when you change variables; otherwise, you are effectively changing the system you are describing. This is why you set $\frac{\partial n}{\partial t} = 0$. Now suppose you change variables to

$$n = \Omega \phi(t) + \sqrt{\Omega}\xi$$

It is important that your states don't change therefore $\xi$ must satisfy

$$ \frac{d \xi}{dt} = -\sqrt{\Omega}\frac{d \phi}{dt}$$

Hope this helps.