Abstract
We discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state Markov chains (continuous time), Onsager theory, and Macroscopic Fluctuation Theory (MFT). For Markov chains, this theory involves a non-linear relation between probability currents and their conjugate forces. Within this framework, we show how the forces can be split into two components, which are orthogonal to each other, in a generalised sense. This splitting allows a decomposition of the pathwise rate function into three terms, which have physical interpretations in terms of dissipation and convergence to equilibrium. Similar decompositions hold for rate functions at level 2 and level 2.5. These results clarify how bounds on entropy production and fluctuation theorems emerge from the underlying dynamical rules. We discuss how these results for Markov chains are related to similar structures within MFT, which describes hydrodynamic limits of such microscopic models.
Highlights
We consider dynamical fluctuations in systems described by Markov chains
We characterise dynamical fluctuations using an approach based on the Onsager–Machlup (OM) theory [36], which is concerned with fluctuations of macroscopic properties of physical systems
We have presented several results for dynamical fluctuations in Markov chains
Summary
We consider dynamical fluctuations in systems described by Markov chains. The nature of such fluctuations in physical systems constrains the mathematical models that can be used to describe them. We characterise dynamical fluctuations using an approach based on the Onsager–Machlup (OM) theory [36], which is concerned with fluctuations of macroscopic properties of physical systems (for example, density or energy). Associated to these fluctuations is a large-deviation principle (LDP), which encodes the probability of rare dynamical trajectories. To describe non-equilibrium processes, that theory must be generalised to include irreversible Markov chains This can be achieved using the canonical structure of fluctuations discovered by Maes and Netocný [38].
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