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We discuss a canonical structure that provides a unifying description ofdynamical large deviations for irreversible finite state Markov chains(continuous time), Onsager theory, and Macroscopic Fluctuation Theory. ForMarkov chains, this theory involves a non-linear relation between probabilitycurrents and their conjugate forces. Within this framework, we show how theforces can be split into two components, which are orthogonal to each other, ina generalised sense. This splitting allows a decomposition of the pathwise ratefunction into three terms, which have physical interpretations in terms ofdissipation and convergence to equilibrium. Similar decompositions hold forrate functions at level 2 and level 2.5. These results clarify how bounds onentropy production and fluctuation theorems emerge from the underlyingdynamical rules. We discuss how these results for Markov chains are related tosimilar structures within Macroscopic Fluctuation Theory, which describeshydrodynamic limits of such microscopic models.c Fluctuation Theory, whichdescribes hydrodynamic limits of such microscopic models.

Canonical structure and orthogonality of forces and currents in irreversible Markov chains