where we have defined
and , according to (22), is given by
A simple algebraic manipulation yields, for :
where . We can then define the rescaled diffusion coefficient:
In certain cases there are conserved quantities in the system that allow to write as a function of . In particular, this is the case for the situation previously discussed in the literature in which the fast reaction is of the form . Then, the original set of evolution equations (1)-(3) read:
so that the quantity remains constant during the whole evolution. Thus, we conclude that also remains constant. Setting and using Eq. (51), which in this case reads , we obtain that . Therefore, the rescaled diffusion coefficient (57) can be rewritten as:
which, in the limit of
reduces to the values obtained in .