Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems

TL;DR

The paper addresses modeling open systems with fluctuating particle numbers by deriving a unifying master equation that encompasses Liouville-like open-system reductions and reaction-diffusion master equations. It introduces a density-based hierarchy $f=(f_0,f_1,\dots)$ evolving under a transport operator $\mathcal{A}_n$ and a coupling operator $\mathcal{Q}_n$, unifying multiple prior approaches under a single mathematical framework. Through detailed sections on Liouville open systems, the Bergman-Lebowitz formulation, and the chemical-diffusion master equation (CDME) with Langevin generalizations, it demonstrates how each existing model is a special case of the general structure and discusses numerical and multiscale implications. The work thus provides a rigorous, versatile foundation for simulating and analyzing complex systems across scales, from molecular to social dynamics, with potential extensions to quantum and data-driven settings. The framework offers practical guidance for consistent reservoir modeling, multiscale coupling, and cross-domain applications where particle number varies in time.

Abstract

A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics.

Figures (3)

• Figure 1: Graphical representation of the open system and associated formalism. The open system $\Omega$, with its boundary surface $\partial\Omega$, is defined as a subsystem of $U$ with $n$ particles. The reservoir is defined as a large system, $U\backslash\Omega=\Omega_{c}$, with $N-n$ particles, since $U$ contains $N$ particles. The $n$ dimensional domain of the phase-space of particles in $\Omega$ is defined as $S^{n}=\mathbb{R}^{3n}\times \Omega^{n}$ while the $N-n$ dimensional domain of the phase-space of particles in $\Omega_{c}$ is defined as $S_{c}^{N-n}=\mathbb{R}^{3(N-n)}\times \Omega^{(n-N)}$. This figure is adapted from Fig.1 of jpa.
• Figure 2: Illustration of particle-based models. a. Change of resolution from molecular to particle resolution, where each molecule is considered as one bead. b. A particle-based reaction-diffusion process for $A+A\rightarrow A$, where $\lambda(y,x_1,x_2)$ is the position dependent rate function. Figure (a) is adapted from Fig.1(a) of jcpor.
• Figure 3: Phase space for a general reaction-diffusion process involving one chemical species. The space $\mathbb{X}$ represents the phase space of one particle $A$ (e.g. $\mathbb{R}^3$), and the distributions described by the chemical diffusion master equation reside in this phase space. Figure adapted from Fig.1(a) of del2024open.