Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An approach to non-equilibrium Markov chains through cycle matrices

Marco Antonio Cruz-de-la-Rosa, Fernando Guerrero-Poblete

Abstract

Analogously to the quantum case considered in Cruz-de-la-Rosa and Guerrero-Poblete (Open Syst. Inf. Dyn. 32, 2550005, 2025), this work proposes a graph-theoretic approach to studying non-equilibrium properties in Markov chains. We prove that the kernel of the incidence matrix associated with the interaction graph of the chain, which consists of cycles, is isomorphic to the space of anti-symmetric matrices with rows sum to zero. The main contribution of this work is the introduction of the called cycle matrices, which constitute a basis for the space of matrices that describe the non-equilibrium.

An approach to non-equilibrium Markov chains through cycle matrices

Abstract

Analogously to the quantum case considered in Cruz-de-la-Rosa and Guerrero-Poblete (Open Syst. Inf. Dyn. 32, 2550005, 2025), this work proposes a graph-theoretic approach to studying non-equilibrium properties in Markov chains. We prove that the kernel of the incidence matrix associated with the interaction graph of the chain, which consists of cycles, is isomorphic to the space of anti-symmetric matrices with rows sum to zero. The main contribution of this work is the introduction of the called cycle matrices, which constitute a basis for the space of matrices that describe the non-equilibrium.
Paper Structure (7 sections, 7 theorems, 43 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Non-equilibrium chains and interaction graph
  3. Interaction graph
  4. $k$-non-equilibrium
  5. Cycle Matrices
  6. Circulant matrices
  7. $1$-non-equilibrium invariant distribution

Key Result

Lemma 1

The matrix $D$ in $(No-BD-Matriz)$ fulfills

Figures (1)

  • Figure 1: Interaction graph.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proof 1
  • Theorem 1
  • Definition 3
  • Proposition 1
  • Proof 2
  • Theorem 2
  • Proof 3
  • ...and 11 more