configuration of the particles when the $n$th particle changes it state from $z_n$ to $z_n^\prime$.

We make the following assumptions on the model:

1.  The graph $(\mathcal{Z}, \mathcal{E})$ is irreducible. \[assm:a1\]

2.  The functions $\lambda_{z,z^\prime}(\cdot)$, $(z,z^\prime) \in \mathcal{E}$, are Lipschitz on $M_1(\mathcal{Z} )$ and there exist positive constants $c, C $ such that $c \leq \lambda_{z,z^\prime}(\xi) \leq C$ for all $(z,z^\prime) \in \mathcal{E}$ and all $\xi \in M_1(\mathcal{Z})$. \[assm:a2\]

Let $D([0, \infty), \mathcal{Z}^N)$ denote the space of $\mathcal{Z}^N$-valued functions on $[0, \infty)$ that are right continuous with left limits (càdlàg), equipped with the Skorohod-$J_1$ topology (see [[@ethier-kurtz Chapter 3]]{}). Note that, under the Lipschitz assumption \[assm:a2\] on the transition rates, the $D([0, \infty), \mathcal{Z}^N)$-valued martingale problem for $\Psi^N$ is well posed (see [[@ethier-kurtz Exercise 15, Section 4.1]]{}); therefore, given an initial configuration of particles $(X_n^N(0), 1 \leq n \leq N) \in \mathcal{Z}^N$, we have a Markov process $\left( (X_n^N(t), 1 \leq n \leq N),  t \geq 0\right) $ on the space $D([0,\infty),\mathcal{Z}^N)$. To describe the process in words, a particle in state $z$ at time $t$ moves to state $z^\prime$ at rate $\lambda_{z,z^\prime}(\mu_N(t))$, independent of everything else; i.e., the evolution of the state of a particle depends on the states 