Reformulating non-equilibrium steady-states and generalised Hopfield discrimination

TL;DR

This paper introduces a thermodynamic reformulation of non-equilibrium steady-state probabilities using minimal paths and entropy, simplifying analysis and extending Hopfield's discrimination model to complex graphs.

Contribution

It presents a novel reformulation of steady-state probabilities based on minimal paths and entropy, providing a thermodynamic interpretation and extending Hopfield's discrimination analysis.

Findings

Steady-state probabilities relate to average exponential of path entropy.

Reformulation generalizes equilibrium statistical mechanics to non-equilibrium systems.

Global synergy from local dissipation enhances discrimination performance.

Abstract

Despite substantial progress in non-equilibrium physics, steady-state (s.s.) probabilities remain intractable to analysis. For a Markov process, s.s. probabilities can be expressed in terms of transition rates using the Matrix-Tree theorem (MTT) in the graph-based linear framework. The MTT reveals that, away from equilibrium, s.s. probabilities become globally dependent on all rates, with expressions growing exponentially in the system size. This overwhelming complexity and lack of thermodynamic interpretation have greatly impeded analysis. Here, we show that s.s. probabilities are proportional to the average of $\exp(-S(P))$, where $S(P)$ is the entropy generated along minimal paths, $P$, in the graph, and the average is taken over a probability distribution on spanning trees. Assuming Arrhenius rates, this "arboreal" distribution becomes Boltzmann-like, with the energy of a tree being its total edge barrier energy. This reformulation offers a thermodynamic interpretation that smoothly generalises equilibrium statistical mechanics and reorganises the expression complexity: the number of distinct minimal-path entropies depends on the entropy production index, a new graph-theoretic measure of non-equilibrium complexity, not on graph size. We demonstrate the power of this reformulation by extending Hopfield's analysis of discrimination by kinetic proofreading to any graph with index 1. We derive a general formula for the error ratio and use it to show that local energy dissipation can yield optimal discrimination through global synergy.