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Irreversibility as divergence from equilibrium

David Andrieux

TL;DR

This work reframes steady-state entropy production as a symmetrized information-theoretic distance to carefully chosen equilibrium dynamics, identifying $P^{m}$ and $P^{e}$ as the closest reversible models to the actual Markov process in the sense of $D(P|Q)$ and $D(Q|P)$. The central result $\tfrac{1}{2}\Delta_i S = D_S(P, P^{x})$ ties irreversibility to a geometric distance from equilibrium, with $P^{m}$ and $P^{e}$ defined via $P^{m}=\tfrac{1}{2}(P+P^*)$ and $P^{e}=s[(P\circ P^*)^{(1/2)}]$, and shown to minimize respective KL divergences. A molecular motor example demonstrates that $D_S(P^{m},P) = D_S(P^{e},P) = \tfrac{1}{2}\Delta_i S$ and recovers the current–affinity form $\Delta_i S = J\,A$, illustrating the practical link between dissipation and information geometry. The paper also develops a geometric picture of the space of Markov dynamics, with equilibrium dynamics forming a two-dimensional manifold and the symmetrized divergence guiding symmetric transport properties, while signaling open questions for time-dependent driving and nonlinear regimes.

Abstract

The entropy production is commonly interpreted as measuring the distance from equilibrium. However, this explanation lacks a rigorous description due to the absence of a natural equilibrium measure. The present analysis formalizes this interpretation by expressing the entropy production of a Markov system as a divergence with respect to particular equilibrium dynamics. These equilibrium dynamics correspond to the closest reversible systems in the information-theoretic sense. This result yields novel links between nonequilibrium thermodynamics and information geometry.

Irreversibility as divergence from equilibrium

TL;DR

This work reframes steady-state entropy production as a symmetrized information-theoretic distance to carefully chosen equilibrium dynamics, identifying and as the closest reversible models to the actual Markov process in the sense of and . The central result ties irreversibility to a geometric distance from equilibrium, with and defined via and , and shown to minimize respective KL divergences. A molecular motor example demonstrates that and recovers the current–affinity form , illustrating the practical link between dissipation and information geometry. The paper also develops a geometric picture of the space of Markov dynamics, with equilibrium dynamics forming a two-dimensional manifold and the symmetrized divergence guiding symmetric transport properties, while signaling open questions for time-dependent driving and nonlinear regimes.

Abstract

The entropy production is commonly interpreted as measuring the distance from equilibrium. However, this explanation lacks a rigorous description due to the absence of a natural equilibrium measure. The present analysis formalizes this interpretation by expressing the entropy production of a Markov system as a divergence with respect to particular equilibrium dynamics. These equilibrium dynamics correspond to the closest reversible systems in the information-theoretic sense. This result yields novel links between nonequilibrium thermodynamics and information geometry.
Paper Structure (5 sections, 26 equations, 1 figure)

This paper contains 5 sections, 26 equations, 1 figure.

Table of Contents

  1. Context and objectives
  2. Time-reversal and entropy production of Markov chains
  3. Irreversibility as divergence from equilibrium
  4. A molecular motor model example
  5. Perspective: Nonequilibrium transport and information geometry

Figures (1)

  • Figure 1: Geometry of the space of Markov chains. The set of equilibrium dynamics is represented as a two-dimensional manifold $\Sigma$. The entropy production is given by the symmetrized KL divergence $D_S$ between $P$ and $P^{e}$ or $P^{m}$, and is obtained by integrating the Fisher information along the $e$-geodesic and the $m$-geodesic (solid lines). By symmetry, the integration can also be performed along the geodesics connecting $P^{e}$ or $P^{m}$ to $P^*$ (dashed lines), leading to the alternative formula $\Delta_i S = 2 D_S(P^x, P^*)$. The KL divergences between $P, P^m,$ and $P^e$ satisfy various inequalities, providing lower bounds for the entropy production (Eqs. (\ref{['ineq1']})-(\ref{['ineq2']}) in appendix).