  The stiff problem is concerned with a thermal conduction model with a
singular barrier of zero volume. In this paper, we shall build the phase
transitions for the stiff problems in one-dimensional space. It turns out that
every phase transition definitely depends on the total thermal resistance of
the barrier, and the three phases correspond to the so-called impermeable
pattern, semi-permeable pattern and permeable pattern of thermal conduction
respectively. For each pattern, the related boundary condition of the flux at
the barrier is also derived. Mathematically, we shall introduce and explore the
so-called snapping out Markov process, which is the probabilistic counterpart
of semi-permeable pattern in the stiff problem.
