# Action Principle and Dynamic Ensemble Theory for Non-equilibrium Markov Chains

## TL;DR

This paper develops a dynamic ensemble theory for non-equilibrium Markov chains based on a minimal free action principle, drawing analogies with equilibrium thermodynamics and providing insights into steady states and entropy production.

## Contribution

It introduces a unified dynamic ensemble framework for non-equilibrium Markov chains using a minimal free action principle, extending thermodynamic concepts to non-equilibrium steady states.

## Key findings

- Dynamic ensemble theory analogous to equilibrium thermodynamics.
- Minimization of free action determines stable non-equilibrium steady states.
- Linear-response theory with reciprocal relations derived from free action approximation.

## Abstract

An overarching action principle, the principle of minimal free action, exists for ergodic Markov chain dynamics. Using this principle and the Detailed Fluctuation Theorem, we construct a dynamic ensemble theory for non-equilibrium steady states (NESS) of Markov chains, which is in full analogy with equilibrium canonical ensemble theory. Concepts such as energy, free energy, Boltzmann macro-sates, entropy, and thermodynamic limit all have their dynamic counterparts. For reversible Markov chains, minimization of Boltzmann free action yields thermal equilibrium states, and hence provide a dynamic justification of the principle of minimal free energy. For irreversible Markov chains, minimization of Boltzmann free action selects the stable NESS, and determines its macroscopic properties, including entropy production. A quadratic approximation of free action leads to linear-response theory with reciprocal relations built-in. Hence, in so much as non-equilibrium phenomena can be modeled as Markov processes, minimal free action serves as a basic principle for both equilibrium and non-equilibrium statistical physics.