  We study the asymptotic behaviour of a gradient system in a regime in which
the driving energy becomes singular. For this system gradient-system
convergence concepts are ineffective. We characterize the limiting behaviour in
a different way, by proving $\Gamma$-convergence of the so-called
energy-dissipation functional, which combines the gradient-system components of
energy and dissipation in a single functional. The $\Gamma$-limit of these
functionals again characterizes a variational evolution, but this limit
functional is not the energy-dissipation functional of any gradient system.
  The system in question describes the diffusion of a particle in a
one-dimensional double-well energy landscape, in the limit of small noise. The
wells have different depth, and in the small-noise limit the process converges
to a Markov process on a two-state system, in which jumps only happen from the
higher to the lower well.
  This transmutation of a gradient system into a variational evolution of
non-gradient type is a model for how many one-directional chemical reactions
emerge as limit of reversible ones. The $\Gamma$-convergence proved in this
paper both identifies the `fate' of the gradient system for these reactions and
the variational structure of the limiting irreversible reactions.
