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

Local detailed balance across scales: from diffusions to jump processes and beyond