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

TL;DR

This paper presents a unified canonical framework for analyzing large deviations, forces, and currents in irreversible Markov chains, revealing orthogonal decompositions and their physical interpretations, and connecting microscopic dynamics to macroscopic fluctuation theories.

Contribution

It introduces a novel orthogonal decomposition of forces in Markov chains and links microscopic dynamics to macroscopic fluctuation theory, enhancing understanding of entropy and dissipation.

Findings

Orthogonal splitting of forces in Markov chains.

Decomposition of rate functions into physically interpretable terms.

Connections established between microscopic Markov models and macroscopic fluctuation theory.

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. 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 Macroscopic Fluctuation Theory, which describes hydrodynamic limits of such microscopic models.c Fluctuation Theory, which describes hydrodynamic limits of such microscopic models.