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and then the temperature expansion, (5.146), is not uniformly valid; in par-
ticular this expansion breaks down when that is, for
Let us therefore rescale write and then equation (5.141)
becomes

with at x = ±1. This branch of the solution, interpreted as a function of is
usually called the hot branch. We seek a solution to give

but we cannot use the boundary conditions here because of the evident non-
uniformity as in equation (5.148). However, if we set at x = 0
(where by symmetry) we obtain

(where we have set and this latter integral can be expressed in terms of
an exponential integral, if that is useful. A solution with the property that
as (which is necessary if matching is to be possible to the solution valid near
x = ±1, where is exponentially small i.e. must satisfy

where as or – 1. (Of course, matching is then
trivial, for we simply choose A = B.) Thus we may integrate (5.149) from x = 0 to,
say, x = 1:

and now we find that increases as increases. So once we have reached this ‘hot
branch’, which is accessed by using the temperature will increase
without bound (or, rather, until some other physics intervenes). Even more details
of this problem, and related thermal processes, can be found in the excellent text on
modelling by Fowler (1997).

The final group of problems bear some relation to those just considered, for they also
involve chemical processes, but we include in this section some mention of biochemical
processes as well.