Long-term behavior of the master equation on a countable network and approximation methods of the (stationary) solutions via finite subsystems in the thermodynamic limit

TL;DR

This work analyzes the master equation on countable networks, establishing existence and uniqueness of solutions in $l^1(\Omega)$ and proving the solution remains a probability vector for irreducible, strongly connected networks. It develops a rigorous thermodynamic-limit framework by introducing finite subnetworks with their own generators $\Gamma^{[F]}$, and demonstrates convergence of finite-subsystem dynamics to the infinite-system dynamics under a suitable generator norm. The paper then characterizes long-time behavior via mean ergodic spaces and shows, under irreducibility and positive recurrence, the time limit exists and converges to a stationary state, with explicit constructions under detailed balance. Through a series of explicit examples (linear chains, hypercubes, and non-equilibrium networks), it clarifies when the time limit and thermodynamic limit commute and when they do not, offering practical guidance for approximating infinite systems by finite subsystems in the thermodynamic limit.

Abstract

The Master equation on directed networks - also called the differential Chapman-Kolmogorov equation - is a linear differential equation, which describes the probability evolution in a discrete system. While this is well understood, if the underlying graph is finite, the mathematics required for the treatment of a network with countable many nodes is way more complicated and advanced. In this paper we provide criteria for the rates of the system, which makes it possible to approximate the solution by finite subsystems in the thermodynamic limit. By writing the phase space as a direct sum of stationary states and states which vanish in the time limit, we give a new proof of when the time limit for an countable, infinite dimensional system exists and when it can be interchanged with the limit of large systems. We give sufficient criteria, when these two limits commute and demonstrate on various examples, what happens when these criteria are violated and only one of these limits exists.

Figures (26)

• Figure 1: Illustrating the time limit and the thermodynamic limit.
• Figure 3: The linear chain with an open end on one side, and a trapping state on the other for $q \in (0, 1)$ and $p \in (\frac{1}{2}, 1)$.
• Figure 4: Illustrating the difference between the original network $\mathcal{S}$ and network $\tilde{S}_n$ associated to the matrix $\mathbb{1} + \frac{t \, \Gamma}{n}$, with the modified link strength and additional self-loops. The walk $(1,1,2,3,3,4) \in \mathcal{WP} (1\xlongrightarrow{5} 4, \, \tilde{S}_n)\subseteq \, \mathcal{W} (1\xlongrightarrow{5} 4, \tilde{S}_n)$, after removing the self-loops, becomes a path, where as the walk $(1,2,3,1,2,3,4) \in \mathcal{W} (1\xlongrightarrow{6} 4, \tilde{S}_n) \backslash \mathcal{WP} (1\xlongrightarrow{6} 4, \tilde{S}_n)$ does not.
• Figure 5: Illustrating the separation of the weight of a walk$\tilde{\omega}\in \mathcal{WP}(1\xlongrightarrow{|\tilde{\omega}|=5}4, \tilde{\mathcal{S}}_n)$ into the weight of a path$\omega\in \mathcal{P}(1\xlongrightarrow{|\omega| = 3}4, \tilde{\mathcal{S}}_n)$ and a product of the weights of self-loops. For a fixed path $\omega = (1,2,3,4) \in \mathcal{P}(1\xlongrightarrow{|\omega|=3}4, \tilde{\mathcal{S}}_{n})$, there are $\binom{|\tilde{\omega}|=5}{|\omega|=3}=10$ many $\tilde{\omega} \in \mathcal{WP}(1\xlongrightarrow{|\tilde{\omega}|=5}4, \tilde{\mathcal{S}}_{n})$ that include the original path, with only additional self-loops, namely $\tilde{\omega}_1 = (1,1,1,2,3,4)$, $\tilde{\omega}_2 = (1,1,2,2,3,4)$, $\tilde{\omega}_3 = (1,1,2,3,3,4)$, $\tilde{\omega}_4 = (1,1,2,3,4,4)$, $\tilde{\omega}_5 = (1,2,2,2,3,4)$, $\tilde{\omega}_6 = (1,2,2,3,3,4)$, $\tilde{\omega}_7 = (1,2,2,3,4,4)$, $\tilde{\omega}_8 = (1,2,3,3,3,4)$, $\tilde{\omega}_9 = (1,2,3,3,4,4)$, and $\tilde{\omega}_{10} = (1,2,3,4,4,4)$.
• Figure 6: The network $\Omega = S \dot{\cup} S^C$ can be decomposed into two disjoint parts $S$ and $S^C$, where there is no link from $S^C$ to $S$. This results in a block-triangular form of both the generator $\Gamma$, as well as the solution operator $\mathrm{e}^{t \, \Gamma}$, making the network reducible.

Theorems & Definitions (113)

• Lemma 1: finite probability flow
• Lemma 2: Boundedness of the solution
• proof
• Lemma 3: Interchanging time derivative and index summation
• proof
• Lemma 4: The solution remains normalized
• proof
• proof
• Lemma 5: The kernel of the generator of an irreducible network
• proof