Communicated by Notices Associate Editor Reza Malek-Madani
Introduction
The starting point of our discussion is the celebrated Cucker-Smale (CS) model, CS07aCS07b, which describes the dynamics of $N$ entities, referred to as agents, with time-dependent positions and velocities $({\mathbf{x}}_i(t),{\mathbf{v}}_i(t)): \mathbb{R}_+ \rightarrow (\Omega ,\mathbb{R}^d)$ governed by
and subject to prescribed initial conditions, $({\mathbf{x}}_i(0),{\mathbf{v}}_i(0))=({\mathbf{x}}_{i0},{\mathbf{v}}_{i0})\in (\Omega ,\mathbb{R}^d)$. The ambient space of positions $\Omega \subset \mathbb{R}^d$ will refer to one of two main scenarios—either $\Omega =\mathbb{R}^d$ or $\Omega =\mathbb{T}^d$. System 1 is a canonical model for alignment dynamics in which pairwise interactions steer towards average heading. Alignment originated in pioneering works Rey87VCBCS95CS07aCS07b, as a key ingredient in self-organization—a unity from within which leads to the emergence of higher-order, large-scale patterns. It is found in ecology—from flocking of birds and schooling of fish to swarming bacteria and insects; in social dynamics of human interactions—from alignment of pedestrians and emerging consensus of opinions to markets and marketing; and in sensor-based networks—from swarming of mobile robots and control of UAVs to macromolecules and metallic rods.
Figure 1.
Flocking of birds.
Figure 2.
Alignment of pedestrians.
The dynamics 1 governs pairwise interactions, $\phi _{ij}(t)\coloneq \phi ({\mathbf{x}}_i(t),{\mathbf{x}}_j(t))$, dictated by a scalar communication kernel, $\phi (\cdot ,\cdot )$ with amplitude $\kappa >0$. We assume that $\phi (\cdot ,\cdot )\in L^\infty (\Omega \times \Omega )$ is a nonnegative symmetric kernel, properly normalized,
The role for the kernel $\phi$ is context-dependent: its approximate shape is either derived empirically, deduced from higher-order principles, learned from the data, or postulated based on phenomenological arguments, e.g., Bal08CFTV10CDMBC07CS07aKTIHC11ST21VZ12 and the references therein. The specific structure of $\phi$, however, is not necessarily known. Instead, we ask how different classes of communication kernels affect the emergent behavior of 1. Here are a few examples for different communication protocols.
A major part of current literature is devoted to the generic class of metric-based kernels, $\phi ({\mathbf{x}},{\mathbf{x}}')=\varphi (|{\mathbf{x}}-{\mathbf{x}}'|)$. Another example is the class of topologically-based kernels, Bal08ST20, where $\phi ({\mathbf{x}},{\mathbf{x}}')=\varphi (\mu ({\mathbf{x}},{\mathbf{x}}'))$ is dictated by the size of the crowd in an intermediate domain of communication ${\mathcal{C}}({\mathbf{x}},{\mathbf{x}}')$ enclosed between ${\mathbf{x}}$ and ${\mathbf{x}}'$,
In particular, if the domain of communication ${\mathcal{C}}$ is shifted to an $R$-ball centered at ${\mathbf{x}}$, one ends up with the nonsymmetric topological kernel MT11$\phi ({\mathbf{x}},{\mathbf{x}}')={\varphi (|{\mathbf{x}}-{\mathbf{x}}'|)}/{\mu (B_R({\mathbf{x}}))}$. A still larger class of pairwise interactions consists of symmetric matrix communication kernels, e.g., ST21. Other important protocols of communication which will not be analyzed here include the class of singular kernels,
in which communication heavily emphasizes near-by neighbors over those farther away, e.g., MMPZ19 and the references therein, and communication based on various random-based protocols found in chemo- and photo-tactic dynamics, the Elo rating system, voter and related opinion-based models, a random-batch method, and consensus-based optimization, to name but a few.Footnote1
1
In view of the limited bibliographic scope available for this article, we refer the reader to Tad21 for a complementary bibliography source.
which are encoded here by a radial potential $U$. A general protocol for rules of engagement, with pairwise interactions driven by alignment, repulsion, and attraction which are dominant in three different zones of proximity, was proposed in Rey87. We shall focus here on the emergent behavior of alignment dynamics, and refer to CFTV10CDMBC07ST21Tad21 and the references therein for results related to more general protocols. To date, we still lack a mathematical theory which analyzes the emergent behavior of the general class of 3Zone models for collective dynamics.
Connectivity
The large-time behavior of 1 depends on the time-dependent weighted graph with agents at the $N$ vertices ${\mathsf{V}}(t)=\{ i\, | \, {\mathbf{x}}_i(t)\}$ and time-dependent edges ${\mathsf{E}}(t)=\{ (i,j) \, | \, i\neq j: \, \phi _{ij}(t) >0\}$. The energy fluctuations associated with 1,
The first equality follows directly from 1 and the assumed symmetry of the adjacency matrix $\Phi (t)=\{\phi _{ij}(t)\}$. The second inequality is a sharp bound in terms of the spectral gap, $\lambda _2(t)\coloneq \lambda _2(\Delta _{\Phi (t)})$, of the graph Laplacian associated with $\Phi (t)$. Here the graph Laplacian, $(\Delta _\Phi )_{\alpha \beta }\coloneq (\sum _{\gamma \neq \alpha } \phi _{\alpha \gamma })\delta _{\alpha \beta }-\phi _{\alpha \beta }(1-\delta _{\alpha \beta })$, and its spectral gap coincide with the Fiedler number which encodes the connectivity properties of the weighted graph of agents $({\mathsf{V}}(t), {\mathsf{E}}(t))$, e.g., CS07a. Indeed, 6 tells us that energy fluctuations are depleted as long as the graph remains (algebraically) connected,
Connectivity, and hence the large time emergence of flocks or swarms, is guaranteed with long-range kernels. In many realistic configurations, however, the communication among “social particles” takes place in local neighborhoods induced by short-range kernels.
Long- and short-range communication kernels will be the topic of the next two sections. Long-range kernels maintain connectivity which in turn imply decay of fluctuations around an emergent cluster. The large-time dynamics with short-range kernels is considerably more complicated—in particular, algebraically connected initial configurations, $\lambda _2(t=0)>0$, may break down into two or more disconnected clusters at a finite time so that $\lambda _2(t_c)=0$. That is, the dynamics with short-range kernels may or may not be stable, which makes it difficult to trace its flocking behavior. Instead, we study here the flocking/swarming behavior with large crowds: large-crowd dynamics tends to stabilize the large-time behavior. As already noted by Immanuel Kant in 1784 “what seems complex and chaotic in the single individual may be seen from the standpoint of the human race as a whole to be a steady and progressive though slow evolution of its original endowment.”
Hydrodynamic description
The large-crowd dynamics of 1 can be encoded in terms of the empirical distribution $f_N(t,{\mathbf{x}},{\mathbf{v}})\coloneq \frac{1}{N}\sum _{i=1}^N \delta _{{\mathbf{x}}_i(t)}({\mathbf{x}})\otimes \delta _{{\mathbf{v}}_i(t)}({\mathbf{v}})$, which is governed by the kinetic equation in state variables $(t,{\mathbf{x}},{\mathbf{v}})\in \mathbb{R}_+\times \Omega \times \mathbb{R}^d$, e.g., HT08HL09CFTV10,
It is driven according to the pairwise communication protocolFootnote2 on the right of 1${}_{2}$,
2
We abbreviate $f=f(t,{\mathbf{x}},{\mathbf{v}}), f'=f(t,{\mathbf{x}}',{\mathbf{v}}')$ and likewise $\square =\square (t,{\mathbf{x}}), \square '=\square (t,{\mathbf{x}}')$, etc.
For $N\gg 1$, the dynamics of $f_N(t,{\mathbf{x}},{\mathbf{v}})$ is captured by its first two moments which we assume to exist—the density $\rho (t,{\mathbf{x}})\coloneq \lim _{N\rightarrow \infty } \int _{\mathbb{R}^d} f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}$, and the momentum $\rho {\mathbf{u}}(t,{\mathbf{x}})\coloneq \lim _{N\rightarrow \infty } \int _{\mathbb{R}^d} {\mathbf{v}}f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}$. They admit the hydrodynamic description in state variable $(t,{\mathbf{x}})\in (\mathbb{R}_+\times \Omega )$,
$$\begin{equation} \left\{\ \ \begin{split} \rho _t + \nabla _{\mathbf{x}}\cdot (\rho {\mathbf{u}}) & = 0, \\ (\rho {\mathbf{u}})_t + \nabla _{\mathbf{x}}\cdot (\rho {\mathbf{u}}\otimes {\mathbf{u}}+ {\mathbf{P}}) &=\kappa {\mathbf{A}}_\phi (\rho ,{\mathbf{u}}). \end{split} \right. \tag{10a}\cssId{texmlid3}{} \end{equation}$$Here, the pressure ${\mathbf{P}}$ on the left of 10a${}_2$ is a symmetric positive-definite stress tensor,$$\begin{equation} \begin{split} {\mathbf{P}}\coloneq \lim _{N\rightarrow \infty }\int ({\mathbf{v}}- {\mathbf{u}})({\mathbf{v}}- {\mathbf{u}})^\top f_N(t,{\mathbf{x}},{\mathbf{v}}) \, \mathrm{d}{\mathbf{v}}, \end{split} \tag{10b}\cssId{texmlid6}{} \end{equation}$$and ${\mathbf{A}}_\phi$ on the right of 10a${}_2$ is the communication protocol associated with $\phi$, corresponding to 9,$$\begin{equation} {\mathbf{A}}_\phi (\rho ,{\mathbf{u}})(t,{\mathbf{x}})\coloneq \int _\Omega \phi ({\mathbf{x}},{\mathbf{x}}')({\mathbf{u}}'-{\mathbf{u}})\rho \rho '\, {\mathsf{{d}}}{\mathbf{x}}'. \tag{10c}\cssId{texmlid5}{} \end{equation}$$
Observe that system 10 is not a purely hydrodynamic description at the macroscopic scale: while the density and velocity, $\rho =\rho (t,{\mathbf{x}})$ and ${\mathbf{u}}={\mathbf{u}}(t,{\mathbf{x}})$, are governed by the macroscopic balances 10a,10c, the pressure in 10b, ${\mathbf{P}}={\mathbf{P}}(t,{\mathbf{x}})$, still requires a closure of the ${\mathbf{v}}$-dependent second-order moments of $f_N$. We recall that in the case of physical particles, one encounters the canonical closure imposed by Maxwellian equilibrium and expressed in terms of the density, velocity, and temperature, $\rho , {\mathbf{u}}$, and $T$,
We mention in this context the special cases of monokinetic closure, ${\mathbf{P}}\equiv 0$, associated with the vanishing temperature $M_{\{\rho ,{\mathbf{u}},T\downarrow 0\}}(t,{\mathbf{x}}) =\rho \delta (|{\mathbf{v}}-{\mathbf{u}}|)$,FK19, as well as examples of an entropic-based closure with measured data and the isothermal closure ${\mathbf{P}}=\rho {\mathbb{I}}_{d\times d}$ (corresponding to constant temperature $T$); see Tad21.
The general case of “social particles,” however, is different: there is no universal closure. The question of closure for the hydrodynamic description of alignment in 10 is therefore left open. We shall revisit this question in the last section of the article. At this stage, we highlight the fact that the decay of energy fluctuations quantified in the next section applies to general mesoscopic pressure stress tensors 10b.
Fluctuations
The total energy of the large-crowd dynamics associated with 1 is given by the second moment (which is assumed to exist)
expressed in terms of the pressure ${\mathbf{P}}$ and the heat flux ${\mathbf{q}}(t,{\mathbf{x}})\coloneq \lim _{N\rightarrow \infty }\frac{1}{2}\int ({\mathbf{v}}-{\mathbf{u}})|{\mathbf{v}}-{\mathbf{u}}|^2f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}$. The energy production on the right of 11 is driven by the enstrophy
The energy can be decomposed as the sum of kinetic and internal energies, $\rho E= \rho e_{{}_{\!K}}+ \rho e$, corresponding to the first two terms in the decomposition of kinetic velocity $\frac{1}{2}|{\mathbf{v}}|^2\equiv \frac{1}{2}|{\mathbf{u}}|^2+\frac{1}{2}|{\mathbf{v}}-{\mathbf{u}}|^2+({\mathbf{v}}-{\mathbf{u}})\cdot {\mathbf{u}}$,
Let $\delta \mathscr{E}(t)$ denote the total energy fluctuations at timeFootnote3$t$,
3
Here and below we abbreviate ${{\mathsf{{d}}}}{\mathsf{m}}_\rho ({\mathbf{x}},{\mathbf{x}}')\coloneq \rho (t,{\mathbf{x}})\rho (t,{\mathbf{x}}')\, {\mathsf{{d}}}{\mathbf{x}}\, {\mathsf{{d}}}{\mathbf{x}}'$.
The first integrand on the right quantifies local fluctuations of macroscopic velocities, ${\mathbf{u}}(t,\cdot )$, while the last two integrands quantify microscopic fluctuations,