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 paper investigates the long-term behavior of the master equation on countable networks, providing criteria for approximating solutions via finite subsystems and analyzing the interchangeability of limits in infinite-dimensional systems.

Contribution

It introduces new criteria for approximating solutions of the master equation on infinite networks and analyzes conditions for the interchange of limits in the thermodynamic regime.

Findings

Criteria established for when finite subsystem approximations are valid.

Conditions identified for the interchangeability of limits in infinite systems.

Examples illustrating the behavior when criteria are not met.

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.