  Systems with conserved currents driven by reservoirs at the boundaries offer
an opportunity for a general analytic study that is unparalleled in more
general out of equilibrium systems. The evolution of coarse-grained variables
is governed by stochastic {\em hydrodynamic} equations in the limit of small
noise.} As such it is amenable to a treatment formally equal to the
semiclassical limit of quantum mechanics, which reduces the problem of finding
the full distribution functions to the solution of a set of Hamiltonian
equations. It is in general not possible to solve such equations explicitly,
but for an interesting set of problems (driven Symmetric Exclusion Process and
Kipnis-Marchioro-Presutti model) it can be done by a sequence of remarkable
changes of variables. We show that at the bottom of this `miracle' is the
surprising fact that these models can be taken through a non-local
transformation into isolated systems satisfying detailed balance, with
probability distribution given by the Gibbs-Boltzmann measure. This procedure
can in fact also be used to obtain an elegant solution of the much simpler
problem of non-interacting particles diffusing in a one-dimensional potential,
again using a transformation that maps the driven problem into an undriven one.
