particle at time $t$ is denoted by $X_n^N(t) \in \mathcal{Z}$, where the superscript $N$ indicates that there are $N$ particles in the system. The empirical measure of the system of particles at time $t$ is defined by $$\begin{aligned}
\mu_N(t) \coloneqq \frac{1}{N} \sum_{n=1}^N \delta_{X_n^N(t)} \in M_1(\mathcal{Z}),\end{aligned}$$ where $\delta_{\cdot}$ denotes the Dirac measure. Here, $M_1(\mathcal{Z})$ denotes the space of probability measures on $\mathcal{Z}$ equipped with the topology of weak convergence. Note that $\mu_N(t)$ is a random measure, i.e., an $M_1(\mathcal{Z})$-valued random variable. Each particle has a set of allowed transitions; to define this, let $(\mathcal{Z}, \mathcal{E})$ be a directed graph with the interpretation that whenever $(z,z^\prime) \in \mathcal{E}$, a particle in state $z$ is allowed to move from $z$ to $z^\prime$. To specify the interaction among particles and evolution of the states of the particles over time, for each $(z,z^\prime) \in \mathcal{E}$, we are given a function $\lambda_{z,z^\prime}: M_1(\mathcal{Z}) \to [0, \infty)$. We consider the generator $\Psi^N$ acting on functions $f$ on $\mathcal{Z}^N$ by

$$\begin{aligned}
\Psi^N f(\mathbf{z}^N) = \sum_{n=1}^N \sum_{z_n^\prime: (z_n, z_n^\prime) \in \mathcal{E}} \lambda_{z_n, z_n^\prime}(\overline{\mathbf{z}^N}) (f(\mathbf{z}^N_{n,z_n,z_n^\prime}) - f(\mathbf{z}^N));\end{aligned}$$

here $\overline{\mathbf{z}^N} = \frac{1}{N}\sum_{n=1}^N \delta_{z_n} \in M_1(\mathcal{Z})$ denotes the empirical measure associated with the configuration $\mathbf{z}^N \in \mathcal{Z}^N$, and $\mathbf{z}^N_{n,z_n,z_n^\prime}$ denotes the resultant 