The study of heterogeneous mean-field systems is a growing area of research. The main motivation is that the homogeneity assumption underlying the classical mean-field models often no longer holds when considering systems outside statistical physics. Therefore, many researchers have studied systems with different heterogeneous assumptions; see, e.g., [1,2,3,4,5,6] and the references therein for an overview of the recent development in the subject. The focus in the current paper is on a particular family of mean-field systems of Gibbs type constructed on block graphs. This family is a particular instance of the models introduced in [2,7] with the particularity lying in the specification of a heterogeneous Gibbs measure as a stationary distribution. This was in particular inspired by the interacting particle systems of Gibbs type on complete graphs analyzed in [8,9].

When studying mean-field systems, we are in particular interested in their asymptotic behavior when the total number of particles N in the system and/or the time t tends to infinity. Classical questions in the study of mean-field systems include their asymptotic behavior when the total number of particles N in the system and/or the time t tends to infinity and under which conditions one can justify the interchangeability of the limits $N\to \infty$ and $t\to \infty$. In particular, we will show for the specific family of Gibbs systems introduced in Section 2 that, under mild assumptions, the associated empirical vector converges weakly and uniformly over compact time intervals, as $N\to \infty$, towards the solution of a McKean–Vlasov system of equations (see Theorem 1). This kind of result is known in the literature as the law of large numbers. Thus, as a consequence, the McKean–Vlasov limiting system can be used to approximate the behavior of the large particles system over finite time intervals. Thence, one might wonder whether or not this approximation is still relevant when time goes to infinity. This question turns out to be related to the stability properties of the limiting McKean–Vlasov system. More precisely, if the latter contains a unique asymptotically stable equilibrium, then the interchangeability of $N\to \infty$ and $t\to \infty$ is fully justified. However, if the McKean–Vlasov system contains multiple equilibria or a unique equilibrium that is not stable, care must be taken since the approximation is not necessarily accurate. The intuition behind this is that if the McKean–Vlasov limit system has several $\omega$-limit sets, the question is which of these sets characterizes the long-time behavior of the system. In addition, in such a case, metastable phenomena might be observed resulting in transitions of the empirical vector process from one $\omega$-limit set to another. For a more detailed discussion, one can consult [7,10] and the references therein. Therefore, the stability of the limiting system is of great interest, which motivated us to investigate it in the current paper.

The classical approach for studying the stability of dynamical systems is through the construction of Lyapunov functions. However, given the nonlinearity of the McKean–Vlasov equations, finding Lyapunov functions in the general case is very challenging. Nevertheless, the Gibbs nature of the systems studied here allows us to adopt the limit of relative entropy approach introduced in [8,9]. The main idea behind this approach takes root in the observation that the relative entropy is a natural Lyapunov function for linear ergodic Markov processes; see, e.g., [11]. Moreover, despite the nonlinearity of the McKean–Vlasov system, the corresponding N-particle processes describe a linear Markov process. Therefore, one then proposes the limit of suitably normalized relative entropies associated with the stationary distribution of the N-particles system as the candidate Lyapunov function, providing that the limit takes a useful form. This approach was shown in [8,9] to be successful for a family of homogeneous mean-field models with jumps of the Gibbs type. The goal of the current paper is to extend the approach to the multi-class setting tackled here.

Notice that for the general non-Gibbs family of mean-field models on block graphs introduced in [2,7], the stationary distribution will often not take an explicit form, and thus a different approach is needed. One possible line of work is the approach proposed in [8] for particular systems with exchangeable particles, where instead of considering limits of relative entropies associated with the stationary distribution, one may consider the large time t and large particles N limits associated with the exchangeable joint probability distribution. Therefore, given the multi-class structure of the systems in [2,7], one might consider specific systems with multi-exchangeable particles. Another possible approach to construct Lyapunov functions is through the Friedlin-Wentzell quasipotential [12]. Indeed, a close tie between the quasipotential associated with small noise stochastic systems and Lyapunov functions for the underlying deterministic models was observed in the literature; see, e.g., [13,14]. Therefore, for the general models introduced in [2,7], one might view the finite N-particles system as a small noise perturbation of the limiting McKean–Vlasov system. Note that the specific quasipotential for these models was introduced in [7]. The idea is then to investigate under which conditions the quasipotential can serve as a Lyapunov function for the McKean–Vlasov system. This, however goes beyond the scope of the current paper.

The rest of the paper is organized as follows: In Section 2, we introduce, the family of Gibbs systems on block graphs and show some preliminary properties. In Section 3, we then prove, the law of large numbers and the convergence of the N-particles empirical vector toward the solution of a McKean–Vlasov system of equations. Therefore, we investigate in Section 4 the local stability of the limiting McKean–Vlasov system by constructing a candidate Lyapunov function. Using a particular Laplace principle associated with the vector of empirical measures (see Proposition 3), we start by computing in Proposition 4 the limit of suitably normalized relative entropies and show that the limit takes an explicit form. Then, we show in Lemma 2 that the limiting function characterizes the fixed points of the McKean–Vlasov system. Finally, Proposition 5 shows that this limiting function satisfies a descent property, which, combined with mild assumptions, shows that it is indeed a local Lyapunov function for the McKean–Vlasov system of equations.

The Gibbs measure concept has a long history and plays an important role in statistical physics. The underlying principle is that, when a system is in equilibrium, states with lower energy are more likely than those with higher energy. Thus, J.W. Gibbs proposed the probability measure $exp\{-U(\mathit{x})/\kappa T\}$ to capture this principle, where $\kappa$ is the Boltzmann constant, T is the temperature, and the function $U\left(\mathit{x}\right)$ gives the energy of the system when it is in the state $\mathit{x}$. Thus, the Gibbs measure, being a model of equilibrium, given an energy function, one might seek a Markov process for which the Gibbs measure is the stationary distribution. Such Markov processes are often referred to as Glauber dynamics thanks to their seminal paper [15]. For a detailed introduction to the topic, one can consult, e.g., [16,17].

Consider a graph $\mathcal{G}=(\mathcal{V},\Xi )$ composed of r blocks $C_1,\dots ,C_r$ of sizes, respectively, $N_1,\dots ,N_r$, where $\mathcal{V}$ is the set of the nodes and $\Xi$ is the set of the edges. Denote by $\left|\mathcal{V}\right|=N_1+\cdots +N_r=N$ the total number of nodes in the graph. Suppose that each block $C_j$ is a clique; that is, all the $N_j$ nodes of the same block are connected. Furthermore, the nodes within the same block $C_j$ are decomposed into two subsets:

The graph $\mathcal{G}=(\mathcal{V},\Xi )$ is thus composed of $2r$ components. This will play an important role in the upcoming analysis. We associate each node of the graph with a particle taking values in a finite state space $\mathcal{Z}=\{1,2,\dots ,K\}\subset \mathbb{N}$. Denoting by $\mathit{x}=(x_n,x_m,n\in C_j^c,m\in C_j^p,1\le j\le r)\in {\mathcal{Z}}^N$ a configuration of the N particles over the graph $\mathcal{G}=(\mathcal{V},\Xi )$, the corresponding local empirical measures describing the state of each component are defined by:

Notice that for each $1\le j\le r$ and $\iota \in \{c,p\}$, the local empirical measure ${\varrho }_z^{j,\iota ,N}\left(\mathit{x}\right)$ takes values in the state spaces ${\mathcal{M}}_1^{N_j^{\iota }}\left(\mathcal{Z}\right)={\mathcal{M}}_1\left(\mathcal{Z}\right)\cap \frac{1}{N_j^{\iota }}{\mathbb{Z}}^K$, where ${\mathcal{M}}_1\left(\mathcal{Z}\right)$ is the space of probability measures on $\mathcal{Z}$ endowed with the topology of weak convergence and $\mathbb{Z}$ is the set of integers. The corresponding empirical vector describing the state of the entire system is denoted by: For ease of readability, in the following, we suppress the dependency of ${\varrho }^{j,\iota ,N}$ and ${\varrho }^N$ upon N. Thus, we simply write ${\varrho }^{j,\iota }$ and $\varrho$ instead. We associate with the configuration $\mathit{x}$ of the N-particles the following energy function: where $V:\mathcal{Z}\to \mathbb{R}$ is the potential function, $W:\mathcal{Z}\times \mathcal{Z}\to \mathbb{R}$ is the symmetric interaction function, and $\beta >0$ is the interaction parameter. Thence, the corresponding Gibbs measure is given by: where $Z_N$ is a normalization constant. We now construct a Markov process, namely a Glauber dynamics, with the Gibbs measure ${\pi }^N$ as its stationary distribution. To this end, let us first introduce the directed graph $(\mathcal{Z},\mathcal{E})$ with $\mathcal{E}\subset \mathcal{Z}\times \mathcal{Z}\backslash \left\{\right(z,z\left)\right|z\in \mathcal{Z}\}$, representing the set of admissible jumps. Thence, whenever $(z,z^{\prime })\in \mathcal{E}$, a particle at state z is allowed to transit to $z^{\prime }\in \mathcal{Z}$ at a rate that depends on the current state of the node and the state of its neighbors. Before going further, suppose the following assumptions throughout the paper.

Define the matrix ${\left(\alpha (z,z^{\prime })\right)}_{z,z^{\prime }\in \mathcal{Z}}$ that identifies the allowed transitions for one particle as: Moreover, let the matrix $A_N(\mathit{x},\mathit{y})$ indexed by the elements $\mathit{x},\mathit{y}\in {\mathcal{Z}}^N$ be defined as: Hence, $A_N$ determines which states of the N-particle system can be reached in one jump. At this stage, there are several ways to construct Glauber dynamics. See, e.g., ([16] [Sect. 3]) for an overview. We propose here the Metropolis dynamic characterized by the following rate matrix: and ${\psi }^N(\mathit{x},\mathit{x})=-{\sum }_{\mathit{y}\ne \mathit{x}}{\psi }^N(\mathit{x},\mathit{y})$, for all $\mathit{x}\in {\mathcal{Z}}^N$. One can verify that the rate matrix ${\psi }^N$ has ${\pi }^N$ as its stationary distribution. In fact, consider two configurations $\mathit{x},\mathit{y}\in {\mathcal{Z}}^N$. By symmetry, one has $A_N(\mathit{x},\mathit{y})=A_N(\mathit{y},\mathit{x})$. Moreover, using (3) and (5), it is easy to check that ${\pi }^N\left(\mathit{x}\right){\psi }^N(\mathit{x},\mathit{y})={\pi }^N\left(\mathit{y}\right){\psi }^N(\mathit{y},\mathit{x})$. Then ${\psi }^N(\mathit{x},\mathit{y})$ satisfies the detailed balance condition with respect to ${\pi }^N$. Hence, ${\pi }^N$ is a stationary distribution of the Markov chain, and the rate matrix ${\psi }^N$ is reversible with respect to ${\pi }^N$. Furthermore, since the graph of allowed transitions $(\mathcal{Z},\mathcal{E})$ is irreducible by Assumption 1, the rate matrix ${\psi }^N$ is also irreducible, and thus, ${\pi }^N$ is the unique stationary distribution.

One might observe from the rate matrix introduced in (5) that the transition between two configurations $\mathit{x}$ and $\mathit{y}$ depends on the difference between their total energy. Therefore, to investigate the large-scale behavior of the system, we first estimate this difference when the total number N of particles in the system is very large. Given the multi-class structure of the system, we take throughout the paper the convention that as $N\to \infty$, $\underset{\begin{array}{c}1\le j\le r\\ \iota \in \{c,p\}\end{array}}{min}{N}_j^{\iota }\to \infty$. In addition, and for the aim of simplicity, we will ignore from now on the environment potential by supposing that $V\equiv 0$ and thus focus on the interaction component of the system. Nevertheless, our results can be easily extended to the case with non-zero potential.

Let us define, for $x,y\in \mathcal{Z}$, $q=(q^{1,c},q^{1,p},\dots ,q^{r,c},q^{r,p})\in {\left({\mathcal{M}}_1\left(\mathcal{Z}\right)\right)}^{2r}$, and $a,b_1,\dots ,b_r\in \mathbb{R}$, the following real-valued functions: and

One might notice from Lemma 1 that if two configurations differ in only one index l, then the jump rates of the Markov process governed by the rate matrix ${\psi }^N$ depend on the states $x_i$ of the other particles $i\ne l$ only through the local empirical measures ${\varrho }^{1,c}\left(\mathit{x}\right),{\varrho }^{1,p}\left(\mathit{x}\right),\dots ,{\varrho }^{r,c}\left(\mathit{x}\right),{\varrho }^{r,p}\left(\mathit{x}\right)$. To emphasize this fact, we introduce the rate functions ${\lambda }^{j,c,N}$ and ${\lambda }^{j,p,N}$ defined, for all $1\le j\le r$, $z,z^{\prime }\in \mathcal{Z}$, $q=(q^{1,c},q^{1,p},\dots ,q^{r,c},q^{r,p})\in {\left({\mathcal{M}}_1\left(\mathcal{Z}\right)\right)}^{2r}$, and $a,b_1,\dots ,b_r\in \mathbb{R}$, by: Thus, when the system is in configuration $\mathit{x}$, a central node $l\in C_j^c$ of a given block $1\le j\le r$ jumps from a state z to $z^{\prime }$ at rate and a peripheral node $l\in C_j^p$ within the block $1\le j\le r$ jumps from a state z to $z^{\prime }$ at rate

Now, using (6) and (7) together with Assumption 1, one can easily prove that, as $N\to \infty$, the functions ${\psi }^{j,c,N}(·,N_j^c,N_j^p)$ and ${\psi }^{j,p,N}(·,N_j^c,N_1^p,\dots ,N_r^p)$ converge, respectively, as $N\to \infty$, toward the functions ${\psi }^c$ and ${\psi }^p$. These functions are defined, for all $1\le j\le r$, $z,z^{\prime }\in \mathcal{Z}$ and $q=(q^{1,c},q^{1,p},\dots ,q^{r,c},q^{r,p})\in {\left({\mathcal{M}}_1\left(\mathcal{Z}\right)\right)}^{2r}$ by: -0.6cm0cm Using this together with the inequalities in (9) and (11), one can further prove that the rate functions ${\lambda }_{z,z^{\prime }}^{j,c,N}(·,N_j^c,N_j^p)$ and ${\lambda }_{z,z^{\prime }}^{j,p,N}(·,N_j^c,N_1^p,\dots ,N_r^p)$ converge, respectively, as $N\to \infty$, toward the functions ${\lambda }_{z,z^{\prime }}^{j,c}$ and ${\lambda }_{z,z^{\prime }}^{j,p}$ defined by:

Fix $q=(q^{1,c},q^{1,p},\dots ,q^{r,c},q^{r,p})\in {\left({\mathcal{M}}_1\left(\mathcal{Z}\right)\right)}^{2r}$; then, for all $1\le j\le r$, the rate matrix $A_q^{j,c}:={({\lambda }_{z,z^{\prime }}^{j,c}\left(q\right))}_{(z,z^{\prime })\in \mathcal{Z}\times \mathcal{Z}}$, with ${\lambda }_{z,z}^{j,c}\left(q\right)=-{\sum }_{z^{\prime }\ne z}{\lambda }_{z,z^{\prime }}^{j,c}\left(q\right)$, is the generator of an ergodic Markov chain with a unique invariant measure defined, for all $z\in \mathcal{Z}$, by: where $Z_N\left(q\right)$ is a normalization constant given by: In fact, this can be easily verified by checking the detailed balance condition. Similarly, the rate matrix $A_q^{j,p}:={({\lambda }_{z,z^{\prime }}^{j,p}\left(q\right))}_{(z,z^{\prime })\in \mathcal{Z}\times \mathcal{Z}}$, with ${\lambda }_{z,z}^p\left(q\right)=-{\sum }_{z^{\prime }\ne z}{\lambda }_{z,z^{\prime }}^{j,p}\left(q\right)$, is the generator of an ergodic Markov chain with a unique stationary distribution defined, for all $z\in \mathcal{Z}$, by: where again $Z_N\left(q\right)$ is the normalization constant given by:

Denote by ${({\mathit{x}}^N\left(t\right))}_{t\ge 0}={(x_n\left(t\right),x_m\left(t\right),n\in C_j^c,m\in C_j^p,1\le j\le r)}_{t\ge 0}$ the stochastic process characterizing the state of the system at each time $t\ge 0$. Hence, ${\left({\mathit{x}}^N\left(t\right)\right)}_{t\ge 0}$ is a Markov process with state space ${\mathcal{Z}}^{2r}$ and transition rate matrix ${\left({\psi }^N(\mathit{x},\mathit{y})\right)}_{\mathit{x},\mathit{y}\in {\mathcal{Z}}^N}$ given by (5). For convenience, we define the following local empirical measures: and let ${\mu }^N\left(t\right)=({\mu }_1^{c,N}\left(t\right),{\mu }_1^{p,N}\left(t\right),\dots ,{\mu }_r^{c,N}\left(t\right),{\mu }_r^{p,N}\left(t\right))$ be the corresponding empirical vector. Observe that ${\mu }_j^{\iota ,N}\left(t\right)(·)={\varrho }_{·}^{j,\iota }\left({\mathit{x}}^N\left(t\right)\right)$. Therefore, at any time $t\ge 0$, a given central (resp. peripheral) particle within the block j jumps from z to $z^{\prime }$, for $z,z^{\prime }\in \mathcal{Z}$, at rate ${\lambda }_{z,z^{\prime }}^{j,c,N}({\mu }^N\left(t\right),N_j^c,N_j^p)$ given in (20) (resp. ${\lambda }_{z,z^{\prime }}^{j,p,N}\left({\mu }^N\left(t\right),N_j^c,N_1^p,\dots ,N_r^p\right)$ given in (21)).

Recall the important notion of multi-exchangeability for multi-class systems in the definition given below.

Given the symmetry of the rate functions within each of the $2r$ classes $C_1^c,C_1^p,\dots ,C_r^c,C_r^p$, a multi-exchangeability assumption made on the initial condition ${\mathit{x}}^N\left(0\right)$ guarantees that, at any $t>0$, ${\mathit{x}}^N\left(t\right)$ is also multi-exchangeable. Therefore, for each $\iota \in \{c,p\}$ and $1\le j\le r$, the Markov processes $x_1,\dots ,x_{N_j^{\iota }}$ characterizing the particles of the class $C_j^{\iota }$ are exchangeable. It follows that the corresponding local empirical measures ${\mu }_j^{\iota ,N}\left(t\right)$ are Markov processes taking values, respectively, in the state spaces ${\mathcal{M}}_1^{N_j^{\iota }}\left(\mathcal{Z}\right)$. For a detailed proof of this claim, one can consult ([18] [Prop. 2.3.3]). The random evolution of the empirical vector ${\mu }^N\left(t\right)$ is summarized as follows.

First, notice that the current setting almost surely allows at most one particle to jump at any given time. Therefore the jumps of ${\mu }^N\left(t\right)$ happen within each component ${\mu }_j^{\iota ,N}\left(t\right)$ at a time and have the form $\frac{1}{N_j^{\iota }}(e_{z^{\prime }}-e_z)$ for $z,z^{\prime }\in \mathcal{Z}$, with $e_z$ representing the unit-coordinate vector in ${\mathbb{R}}^K$ in the z-direction. Moreover, if ${\mu }^N\left(t\right)=q=(q^{1,c},q^{1,p},\dots ,q^{r,c},q^{r,p})\in {\prod }_{j=1}^r{\prod }_{\iota \in \{c,p\}}{\mathcal{M}}_1^{N_j^{\iota }}\left(\mathcal{Z}\right)$ at time $t\ge 0$, then there are $N_j^{\iota }{q}_z^{j,\iota }$ particles of the class $C_j^{\iota }$ at each state $z\in \mathcal{Z}$. Each of these particles jumps independently to $z^{\prime }\in \mathcal{Z}$ at rate ${\lambda }_{z,z^{\prime }}^{j,c,N}(q,N_j^c,N_j^p)$ for $\iota =c$ and ${\lambda }_{z,z^{\prime }}^{j,p,N}(q,N_j^c,N_1^p,\dots ,N_r^p)$ for $\iota =p$. Therefore, the local central empirical processes ${\mu }_j^{c,N}\left(t\right)$ transit from $q^{j,c}$ to $q^{j,c}+\frac{1}{N_j^c}(e_{z^{\prime }}-e_z)$ at rate $N_j^c{q}_z^{j,c}{\lambda }_{z,z^{\prime }}^{j,c,N}(q,N_j^c,N_j^p)$. Similarly, the peripheral local empirical processes ${\mu }_j^{p,N}\left(t\right)$ transit from $q^{j,p}$ to $q^{j,p}+\frac{1}{N_j^p}(e_{z^{\prime }}-e_z)$ at rate $N_j^p{q}_z^{j,p}{\lambda }_{z,z^{\prime }}^{j,p,N}(q,N_j^c,N_1^p,\dots ,N_r^p)$. Thence, the infinitesimal generator associated with the empirical vector ${\mu }^N\left(t\right)$ is given for any real-valued function $\mathsf{\Phi }$ on ${\left({\mathcal{M}}_1\left(\mathcal{Z}\right)\right)}^{2r}$ by: One might also describe the changes in the system locally by introducing the infinitesimal generators corresponding to the classes $C_j^{\iota }$ and defined for functions f on ${\mathcal{M}}_1\left(\mathcal{Z}\right)$ by: