Matrix approach to generalized ensemble theory for nonequilibrium discrete systems

TL;DR

This work introduces a universal matrix-based generalized ensemble for nonequilibrium discrete systems, representing any probability distribution as a generalized Boltzmann form through an observation matrix $\mathbb A$ and a Boltzmann vector $\bm B$. It provides a rigorous algebraic framework that yields nonequilibrium thermodynamic relations and fluctuation-dissipation relations, without requiring ergodicity or detailed balance. The theory clarifies gauge freedom, identifies minimal sufficient statistics via minKL inference with a reference distribution $\bm Q$, and shows how classical ensembles and Tsallis statistics fit within the same formalism. A key case study on Markov jump processes demonstrates how dynamical data alongside the matrix representation distinguishes equilibrium from nonequilibrium steady states and yields exact FDRs. Overall, the framework offers a principled, scalable approach to quantify thermodynamics and responses in far-from-equilibrium discrete systems with broad potential applications.

Abstract

A universal and rigorous ensemble framework for nonequilibrium system remains lacking. Here, we provide a concise framework for the generalized ensemble theory of nonequilibrium discrete systems using matrix-based approach. By introducing an observation matrix, we show that any discrete probability distribution can be formulated as a generalized Boltzmann distribution, with observables and their conjugate variables serving as basis vectors and coordinates in a vector space. Within this framework, we identify the minimal sufficient statistics required to infer the Boltzmann distribution. The nonequilibrium thermodynamic relations and fluctuation-dissipation relations naturally emerge from this framework. Our findings provide a new approach to developing generalized ensemble theory for nonequilibrium discrete systems.

Figures (2)

• Figure 1: The illustration of matrix representation of 3-spin model with binary state ($\pm1$). Spin microstates are ordered from spin 3 to spin 1. The left column shows observables evaluated for each microstate, such as $s_2 s_1$, which denotes the product of spins 2 and 1. Each configuration includes a complete set of observables ranging from single-spin terms to higher-order products, ultimately forming the Hadamard matrix. The first row is the normalized vector.
• Figure 2: For a system with three microstates $\{ \sigma_1, \sigma_2, \sigma_3 \}$, assuming that $\bm{a_1}$ is orthogonal to the subspace $\mathcal{V}=\text{span}\{\bm{a_2},\bm{a_3}\}$. $b_2$ and $b_3$ serve as coordinates on this plane, while $b_1$ is determined as a function of $b_2$ and $b_3$, forming a curved surface. The reference distribution $\bm{Q}$ and the target distribution $\bm{P}$, as labeled on the surface, can be projected onto the plane $\mathcal{V}$. The vector $\bm{L}$, defined as the difference between mapped points, can be expressed in terms of $\bm{a_2}$ and $\bm{a_3}$.