  Diffusive dynamics in presence of deep energy minima and weak nongradient
forces can be coarse-grained into a mesoscopic jump process over the various
basins of attraction. Combining standard weak-noise results with a path
integral expansion around equilibrium, we show that the emerging transition
rates satisfy local detailed balance (LDB). Namely, the log ratio of the
transition rates between nearby basins of attractions equals the free-energy
variation appearing at equilibrium, supplemented by the work done by the
nonconservative forces along the typical transition path. When the mesoscopic
dynamics possesses a large-size deterministic limit, it can be further reduced
to a jump process over macroscopic states satisfying LDB. The persistence of
LDB under coarse graining of weakly nonequilibrium states is a generic
consequence of the fact that only dissipative effects matter close to
equilibrium.
