Swarming: hydrodynamic alignment with pressure

Abstract

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering toward a weighted average heading. We consider the class of so-called $p$-alignment hydrodynamics, based on $2p$-Laplacians and weighted by a general family of symmetric communication kernels. The main new aspect here is the long-time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as “entropic”, and prove the flocking of $p$-alignment hydrodynamics, driven by singular kernels with a general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.

1. Introduction—Alignment dynamics and entropic pressure

Alignment reflects steering toward an average heading Rey1987. It plays an indispensable role in the process of emergence in swarming dynamics and, in particular—in flocking, herding, schooling, …, VCBCS1995CF2003CKFL2005CS2007aCS2007bBal2008Kar2008VZ2012MCEB2015PT2017, as well as the formation of other self-organized clustering in human interactions and in dynamics of sensor-based networks Kra2000BeN2005BHT2009JJ2015RDW2018DTW2019Alb2019; more can be found in MT2014, §9, in the book series on active matter BDT2017/19BCT2022, and in the recent Gibbs’ Lecture Tad2022a.

We discuss alignment dynamics in two parallel descriptions. Historically, alignment models were introduced in the context of agent-based description Aok1982Rey1987VCBCS1995. In particular, our discussion is motivated by the celebrated Cucker–Smale model CS2007aCS2007b, in which alignment is governed by weighted graph Laplacians. Our main focus, however, is on the corresponding hydrodynamic description, the so-called Euler alignment equations, governed by a general class of weighted $p$-graph Laplacians HT2008CFTV2010HHK2010Shv2021. In both cases—the agent-based and hydrodynamic descriptions—the weights for the protocol of alignment reflect pairwise interactions, and they are quantified by proper communication kernel. Communication kernels are either derived empirically, deduced from higher-order principles, learned from the data, or postulated based on phenomenological arguments; e.g., CS2007aCDMBC2007Bal2008Ka2011GWBL2012JJ2015LZTM2019MLK2019ST2020b. The specific structure of such kernels, however, is not necessarily known. Instead, we ask how different classes of communication kernels affect the swarming behavior.

The passage from agent-based to hydrodynamic descriptions requires a proper notion of hydrodynamic pressure. In Section 1 we introduce a class of entropic pressures for hydrodynamic alignment, and in Section 2 we extend the discussion to the larger class of hydrodynamic $p$-alignment. Our goal is to make a systematic study of the long-time swarming behavior of hydrodynamic alignment, portrayed in Section 3, with entropic pressure laws. Specifically, we use the decay of energy fluctuations, discussed in Section 4, in order to quantify the emergence of flocking behavior, depending on the communication kernel. Almost all available literature is devoted to the case of pressureless alignment. We review these results in Section 5. The main theme here is unconditional flocking for pressureless $p$-alignment, driven by heavy-tailed communication kernels. In Section 6 we discuss hydrodynamic alignment driven by a general class of entropic pressure. The remarkable aspect here is that despite the lack of closure of such entropic pressure laws, there holds unconditional flocking of $p$-alignment driven by singular-head, heavy-tailed communication kernels. We are aware that the methodology developed here can be utilized with other Eulerian-based dissipative systems. The detailed computations are outlined in Appendices A, B, C, D, and E.

1.1. Hydrodynamic description of alignment

We study the long-time behavior of the hydrodynamic description for alignment,

$$\begin{equation} \begin{cases} & \partial _{t}\rho +\nabla _{\mathbf{x}}\cdot (\rho {\mathbf{u}}) = 0, \\ & \partial _{t}(\rho {\mathbf{u}})+\nabla _{\mathbf{x}}\cdot (\rho {\mathbf{u}}\otimes {\mathbf{u}}+\mathbb{P}) = {\mathbf{A}}(\rho ,{\mathbf{u}}), \end{cases} \quad (t,{\mathbf{x}})\in ({\mathbb{R}}_t, {\mathbb{R}}^d). {\tag{1.1a}} \end{equation}$$The dynamics is captured by density $\rho : {\mathbb{R}}_t\times {\mathbb{R}}^d \mapsto {\mathbb{R}}_+$, momentum $\rho {\mathbf{u}}: {\mathbb{R}}_t\times {\mathbb{R}}^d \mapsto {\mathbb{R}}^d$, and pressure tensor $\mathbb{P}: {\mathbb{R}}_t\times {\mathbb{R}}^d \mapsto {\mathbb{R}}^d\times {\mathbb{R}}^d$, subject to initial data $\displaystyle (\rho ,{\mathbf{u}},\mathbb{P})_{|_{t=0}} = (\rho _{0},{\mathbf{u}}_{0},\mathbb{P}_{0})$, and is driven by an alignment term acting on the support ${\mathcal{S}}(t)\coloneq \operatorname {supp}\rho (t,\cdot )$,$$\begin{multline} {\mathbf{A}}(\rho ,{\mathbf{u}}) \coloneq \displaystyle {\int \limits _{{\mathcal{S}}(t)}}\phi ({\mathbf{x}},{\mathbf{x}'})({\mathbf{u}}(t,{\mathbf{x}'})-{\mathbf{u}}(t,{\mathbf{x}}))\rho (t,{\mathbf{x}})\rho (t,{\mathbf{x}'})\, \mathrm{d}{\mathbf{x}'}, \\ \phi ({\mathbf{x}},{\mathbf{x}'})=\phi ({\mathbf{x}'},{\mathbf{x}}). {\tag{1.1b}} \end{multline}$$The alignment term on the right reflects steering toward an average heading. Here, different weighted averages are dictated by symmetric communication kernel $\phi (\cdot ,\cdot )$. Prototypical examples include metric kernels $\phi ({\mathbf{x}},{\mathbf{x}'})=k(|{\mathbf{x}}-{\mathbf{x}'}|)$, which go back to CS2007a. Other classes of symmetric kernels that are either dictated by the problem or learned from the data can be found in GWBL2012JJ2015LZTM2019MLK2019, and finally we mention topologically based kernels studied in ST2020b $\phi ({\mathbf{x}},{\mathbf{x}'})=k(m(C({\mathbf{x}},{\mathbf{x}'})))$, where $\displaystyle m(C({\mathbf{x}},{\mathbf{x}'}))= \int _{C}\rho (t,{\mathbf{z}})\, \mathrm{d}{\mathbf{z}}$ is the mass enclosed in an intermediate domain $C=C({\mathbf{x}},{\mathbf{x}'})$ with tips at ${\mathbf{x}}$ and ${\mathbf{x}'}$. The prominent role of metric kernels enters when we assume that there exists a radial kernel, $k(r)$, such that$$\begin{equation} \phi ({\mathbf{x}},{\mathbf{x}'})\geqslant k (|{\mathbf{x}}-{\mathbf{x}'}|). {\tag{1.1c}} \end{equation}$$

We further assume that the metric kernel $k (r)$ is decreasing with the distance $r$, reflecting the typical observation that the intensity of alignment decreases with the distance. In particular, we address general metric kernels $\phi (|\cdot |)$ whether decreasing or not, in terms of their decreasing envelope $k (r)\coloneq \min \{\phi (|{\mathbf{x}}|) \ | \ |{\mathbf{x}}|\leqslant r\}$. Observe that we do not place any restriction on the upper bound of $\phi$; in particular, therefore, our discussion includes the important subclass of singular communication kernels $k(r)=r^{-\alpha }, \ \alpha >0$ ST2017aDKRT2018MMPZ2019AC2021b.

1.2. Entropic pressure

System 1.1 is not closed in the sense that the pressure $\mathbb{P}$ is not specified—neither in terms of algebraic relations with $(\rho ,{\mathbf{u}})$, nor do we specify the precise dynamics of $\mathbb{P}$. We do not dwell here on the details of the underlying pressure tensor. Instead, we treat a rather general class of pressure laws satisfying an essential structural (dissipative) property which, as we shall show, maintains long-time flocking behavior. This brings us to the following.

Definition 1.1 (Entropic pressure).

We say that $\mathbb{P}$ is an entropic pressure associated with 1.1 if it has a nonnegative trace, $\rho e{}_{_{\mathbb{P}}}\coloneq \frac{1}{2} \operatorname {trace}(\mathbb{P})\geqslant 0$, which satisfies

$$\begin{equation} \begin{split} &\partial _{t}(\rho e{}_{_{\mathbb{P}}} )+\nabla _{{\mathbf{x}}}\cdot (\rho e{}_{_{\mathbb{P}}} {\mathbf{u}} +\mathbf{q} )+\operatorname {trace}(\mathbb{P}\nabla {\mathbf{u}})\\ &\qquad \leqslant - 2\int \limits _{{\mathcal{S}}(t)} \phi ({\mathbf{x}},{\mathbf{x}'})e{}_{_{\mathbb{P}}}(t,{\mathbf{x}})\rho (t,{\mathbf{x}})\rho (t,{\mathbf{x}'}) \, \mathrm{d}{\mathbf{x}'}. \end{split}{\tag{1.2}} \end{equation}$$

Here $\mathbf{q}$ is an arbitrary $C^1$-flux.

Why entropic pressure?

System 1.1 falls under the general category of hyperbolic balance laws Daf2016, Chapter III, and 1.2 can be viewed as an entropy inequality associated with such balance law. To this end, we note that a formal manipulation of the mass and momentum equations, 1.1a$\displaystyle {}_1 \times \frac{|{\mathbf{u}}|^2}{2} +$ 1.1a${}_2 \cdot {\mathbf{u}}$ yields⁠¹

¹

Here and below for a quantity $\square =\square (t,{\mathbf{x}})$ we abbreviate $\square '\coloneq \square (t,{\mathbf{x}'})$.

✖

$$\begin{equation} \begin{split} &\partial _t \Big (\frac{\rho }{2}|{\mathbf{u}}|^2\Big ) + \nabla _{\mathbf{x}}\cdot \Big (\frac{ \rho }{2}|{\mathbf{u}}|^2{\mathbf{u}} +\mathbb{P}{\mathbf{u}}\Big ) - \operatorname {trace} \big (\mathbb{P} \nabla {\mathbf{u}}\big ) \\ & \qquad = - \int \limits _{{\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\rho {\mathbf{u}}\cdot ({\mathbf{u}}-{\mathbf{u}}')\rho ' \, \mathrm{d}{\mathbf{x}'}. \end{split}{\tag{1.3}} \end{equation}$$

Adding the entropic description of the pressure postulated in 1.2 leads to the entropic statement for the total energy, $\displaystyle E\coloneq \frac{|{\mathbf{u}}|^2}{2} +e{}_{_{\mathbb{P}}}$,

$$\begin{equation} \partial _t (\rho E) + \nabla _{\mathbf{x}}\cdot (\rho E{\mathbf{u}}+\mathbb{P}{\mathbf{u}}+\mathbf{q}) \leqslant -\int \limits _{{\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\big (|{\mathbf{u}}|^2-{\mathbf{u}}\cdot {\mathbf{u}}'+2e{}_{_{\mathbb{P}}}\big )\rho \rho ' \, \mathrm{d}{\mathbf{x}'}. {\tag{1.4}} \end{equation}$$

Thus, the notion of entropic pressure 1.2 complements the balance laws in 1.1 to form the entropy inequality 1.4.

To further motivate this notion of entropic pressure, we appeal to its underlying kinetic formulation. The hydrodynamics 1.1 corresponds to the large-crowd dynamics of $N$ agents with position/velocity $\displaystyle ({\mathbf{x}}_{i}(t),{\mathbf{v}}_{i}(t)): {\mathbb{R}}_t\mapsto {\mathbb{R}}^{d}\times {\mathbb{R}}^{d}$, governed by the celebrated agent-based alignment model of Cucker and Smale CS2007aCS2007b,

$$\begin{equation} \begin{cases} \displaystyle \frac{\mathrm{d}}{\mathrm{d} t}{\mathbf{x}}_{i}(t) = {\mathbf{v}}_{i}(t), \\ \displaystyle \frac{\mathrm{d}}{\mathrm{d} t} {\mathbf{v}}_{i}(t) = \frac{1}{N}\sum _{j = 1}^{N}\phi _{ij}(t)({\mathbf{v}}_{j}(t)-{\mathbf{v}}_{i}(t)), \end{cases} \qquad i=1,2,\ldots , N. {\tag{1.5}} \end{equation}$$

The alignment dynamics is driven by a weighted graph Laplacian on the right of 1.5${}_2$, dictated by the symmetric communication kernel, $\phi _{ij}(t)\coloneq \phi ({\mathbf{x}}_{i}(t),{\mathbf{x}}_{j}(t))$. The passage from the agent-based to the hydrodynamic description is realized by moments of the empirical distribution

$$\begin{equation*} f_N(t,{\mathbf{x}},{\mathbf{v}}) \coloneq \frac{1}{N }\sum _{i=1}^{N }\delta _{{\mathbf{x}}_{i}(t)}\otimes \delta _{{\mathbf{v}}_{i}(t)}, \qquad (t,{\mathbf{x}},{\mathbf{v}})\in {\mathbb{R}}_{t}\times {\mathbb{R}}^d\times {\mathbb{R}}^{d}. \end{equation*}$$

The large-crowd limits, which are assumed to exist, recover 1.1 with

$$\begin{equation*} \rho (t,{\mathbf{x}})= \lim _{N \rightarrow \infty }\int \limits _{{\mathbb{R}}^d} f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}} \quad \text{and}\quad \rho {\mathbf{u}}(t,{\mathbf{x}}) = \lim _{N \rightarrow \infty }\int \limits _{{\mathbb{R}}^d} {\mathbf{v}} f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}. \end{equation*}$$

This passage from agent-based to macroscopic description is outlined in Appendix A.1. It was justified for smooth kernels HT2008CFTV2010CCR2011FK2019NP2021Shv2021 and at least mildly singular kernels Pes2015PS2019MMPZ2019. In this context, the pressure or Reynolds stress tensor corresponds to the second-order moments

$$\begin{equation} \mathbb{P}(t,{\mathbf{x}}) = \lim _{N \rightarrow \infty }\int \limits _{{\mathbb{R}}^d} ({\mathbf{v}}-{\mathbf{u}})\otimes ({\mathbf{v}}-{\mathbf{u}})f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}. {\tag{1.6}} \end{equation}$$

We observe that the kinetic description of pressure in 1.6 is consistent with the entropic inequality postulated in 1.2. Indeed, $\rho e{}_{_{\mathbb{P}}}\coloneq \frac{1}{2} \operatorname {trace}(\mathbb{P})$ is the internal energy which quantifies microscopic fluctuations around the bulk velocity ${\mathbf{u}}$,

$$\begin{equation} \rho e{}_{_{\mathbb{P}}} = \lim _{N \rightarrow \infty } \int \limits _{{\mathbb{R}}^d} \frac{1}{2} |{\mathbf{v}}-{\mathbf{u}}|^2f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}. {\tag{1.7}} \end{equation}$$

This kinetic description of internal energy yields (detailed derivation is carried out in Appendix A.2),

$$\begin{equation} \partial _{t}(\rho e{}_{_{\mathbb{P}}} )+\nabla _{{\mathbf{x}}}\cdot (\rho e{}_{_{\mathbb{P}}} {\mathbf{u}} +\mathbf{q}_h )+\operatorname {trace}(\mathbb{P}\nabla {\mathbf{u}}) = - 2 \int \limits _{{\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})e{}_{_{\mathbb{P}}}(t,{\mathbf{x}})\rho \rho ' \, \mathrm{d}{\mathbf{x}'}, {\tag{1.8}} \end{equation}$$

with the so-called heat flux $\displaystyle \mathbf{q}_h\coloneq \lim _{N \rightarrow \infty }\frac{1}{2}\int |{\mathbf{v}}-{\mathbf{u}}|^2({\mathbf{v}}-{\mathbf{u}})f_N(t,{\mathbf{x}},{\mathbf{v}})\, \mathrm{d}{\mathbf{v}}$. Formally, any kinetic-based pressure tensor is in particular an entropic pressure, in the sense of satisfying the equality 1.8. But here one encounters the familiar problem of lack of closure, which arises whenever one is dealing with the highest truncated ${\mathbf{v}}$-moments of $f_N$: the second moments encoded in $\rho e{}_{_{\mathbb{P}}}$ and $\mathbb{P}$ now require the third moment encoded in $\mathbf{q}_h$, and so on. In classical particle dynamics, the closure problem is resolved by compatibility with a preferred state of thermal equilibrium, a Maxwellian induced by the thermal equilibrium of the system Lev1996Gol1998Cer2003Vil2003. In the current setup, however, the agent-based dynamics 1.5 governs active matter made of social particles which admit no universal Maxwellian closure. Then, there are multiple reasons which led us to postulate the corresponding entropy inequality 1.2.

Scalar pressure

We discuss the case of scalar pressure law $\displaystyle {\mathbb{P}}={\scriptstyle \mathbb{P}}{\mathbb{I}}$. A large part of the existing literature on swarming assumes a mono-kinetic closure,

$$\begin{equation} f_N(t,{\mathbf{x}},{\mathbf{v}}) \stackrel{N\rightarrow \infty }{\longrightarrow } \rho (t,{\mathbf{x}})\delta ({\mathbf{v}}-{\mathbf{u}}(t,{\mathbf{x}})), {\tag{1.9}} \end{equation}$$

which is realized in terms of zero pressure, ${\scriptstyle \mathbb{P}}=0$; e.g., HT2008CFTV2010FK2019NP2021Shv2021 and the references therein. We mention the derivation from first principles Bia2012, the isentropic closure, ${\scriptstyle \mathbb{P}}=\rho ^\gamma$, of KMT2013KMT2015KV2015Cho2019TCGW2020Shv2022, or equations of state fitted by observation that can be found in Sin2021 as examples of detailed thermodynamic closures for scalar pressure laws in the form of equality in 1.10.

The notion of entropic pressure covers all these scalar examples of entropic pressure laws, as it applies to a broad class of pressure laws satisfying the entropy inequality postulated in 1.2 but otherwise require no algebraic closure. Indeed, our notion of entropic pressure becomes more transparent in the scalar case $\mathbb{P}={\scriptstyle \mathbb{P}}{\mathbb{I}}$, where the inequality postulated in 1.2 for $\displaystyle {\scriptstyle \mathbb{P}}\coloneq \frac{2}{d}\rho e{}_{_{\mathbb{P}}}$ reads (assuming no heat flux $\mathbf{q}=0$),

$$\begin{equation} \partial _t {\scriptstyle \mathbb{P}}+ \nabla _{\mathbf{x}}\cdot ({\scriptstyle \mathbb{P}}{\mathbf{u}})+\frac{2}{d}{\scriptstyle \mathbb{P}}\nabla _{\mathbf{x}}\cdot {\mathbf{u}} \leqslant -2{\scriptstyle \mathbb{P}}\int \limits _{{\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\rho (t,{\mathbf{x}'})\, \mathrm{d}{\mathbf{x}'}. {\tag{1.10}} \end{equation}$$

Formal manipulation, 1.10$\times \rho ^{-\gamma } -$ 1.1a$\displaystyle {}_1 \times \gamma \rho ^{-\gamma -1}{\scriptstyle \mathbb{P}}$ with $\gamma =1+\frac{2}{d}$, leads to the equivalent entropic statement for $S=\ln \big ({\scriptstyle \mathbb{P}}\rho ^{-\gamma }\big )$,

$$\begin{equation} \begin{split} &\partial _t (\rho S) + \nabla _{\mathbf{x}}\cdot (\rho {\mathbf{u}} S) \\ &\qquad \leqslant -2\int \limits _{{\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\rho (t,{\mathbf{x}})\rho (t,{\mathbf{x}'})\, \mathrm{d}{\mathbf{x}'}, \qquad S\coloneq \ln \big ({\scriptstyle \mathbb{P}}\rho ^{-(1+\frac{2}{d})}\big ). \end{split}{\tag{1.11}} \end{equation}$$

We point out that the inequality 1.11 is the reversed entropy inequality encountered for $-S$ in compressible Euler equations. The difference, which was already noted in HT2008, §6, is due to different states of thermodynamic equilibria.

Entropic energy dissipation

An entropy inequality is intimately connected with the irreversibility of the underlying process; see, e.g., the enlightening discussion in Vil2003, §2.4. In the present context of hydrodynamic alignment, the entropy inequality 1.2, or in its equivalent form 1.4, yields

$$\begin{equation} \begin{split} \frac{\mathrm{d}}{\mathrm{d} t} \int \limits _{{\mathcal{S}}(t)}&\rho E\, \mathrm{d}{\mathbf{x}} + \int \limits _{\partial {\mathcal{S}}(t)} \Big ((\mathbb{P}{\mathbf{u}})\cdot {\mathbf{n}} +\mathbf{q}\cdot {\mathbf{n}}\Big ){\mathrm{d}}S \\ & \leqslant - \iint \limits _{{\mathcal{S}}(t)\times {\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\big (|{\mathbf{u}}|^2-{\mathbf{u}}\cdot {\mathbf{u}}' +2 e{}_{_{\mathbb{P}}}\big )\rho \rho ' \, \mathrm{d}{\mathbf{x}}\, \mathrm{d}{\mathbf{x}'} \\ & =- \frac{1}{2}\iint \limits _{{\mathcal{S}}(t)\times {\mathcal{S}}(t)}\phi ({\mathbf{x}},{\mathbf{x}'})\Big (|{\mathbf{u}}'-{\mathbf{u}}|^2 +2e{}_{_{\mathbb{P}}}+2e'_{_{\mathbb{P}}}\Big )\rho \rho ' \, \mathrm{d}{\mathbf{x}}\, \mathrm{d}{\mathbf{x}'} <0, \end{split} {\tag{1.12}} \end{equation}$$

which reflects the dissipativity of the total energy $\displaystyle \int \rho E\, \mathrm{d}{\mathbf{x}}$. Thus, the entropy inequality 1.2 complements the balance laws in 1.1 to govern the energy dissipation 1.12. This is reminiscent of P.-L. Lions’s notion of dissipative solutions in the context of the Euler equations Lio1996, §4.4.

One of the main aspects of this work is dealing with arbitrary pressure, without any specifics about the second-order closure for $\mathbb{P}$. The definition of entropic pressure in 1.2 is not concerned with the detailed balance of internal energy. Instead, its main purpose is to secure the dissipative nature of the total energy, $\displaystyle \rho E$. This partially echoes Vicsek and Zaferis, who argued that in the context of collective motion “The source of energy making the motion possible …are not relevant” VZ2012, §1.1. Here, we abandon a closure in the form of thermal equality 1.8 and, instead, retain the inequality postulated in 1.2, compatible with the dissipativity of internal fluctuations, which we argued for in Tad2021, p. 501. In particular, our definition of a pressure in 1.2 can be realized in any intermediate scale between the microscopic agent-based description, 1.5, and the macroscopic hydrodynamics 1.1, and hence can be viewed as mesoscopic. These considerations become even more pronounced when we extend our discussion to a larger class of so-called $p$-alignment hydrodynamics.

2. $p$-alignment

We begin with the agent-based description,

$$\begin{equation} \begin{cases} \displaystyle \frac{\mathrm{d}}{\mathrm{d} t}{\mathbf{x}}_{i}(t) = {\mathbf{v}}_{i}(t), \\ \displaystyle \frac{\mathrm{d}}{\mathrm{d} t} {\mathbf{v}}_{i}(t) = \frac{1}{N}\sum _{j = 1}^{N}\phi _{ij}(t)|{\mathbf{v}}_{j}(t)-{\mathbf{v}}_{i}(t)|^{2p-2}({\mathbf{v}}_{j}(t)-{\mathbf{v}}_{i}(t)), \end{cases} \quad i=1,2,\ldots , N. {\tag{2.1}} \end{equation}$$

The case $p=1$ coincides with the Cucker–Smale model 1.5, while for $p>1$, the alignment term on the right of 2.1 corresponds to the weighted graph $2p$-Laplacian⁠² which is found in recent applications of neural networks FZN2021, spectral clustering BH2009, and semi-supervised learning ST2019Fu2021. In the context of alignment dynamics it was introduced in HHK2010CCH2014. We were motivated by the example of the Elo rating system JJ2015DTW2019, in which the alignment of scalar ratings $\{q_i\}$ is governed by the odd function of local gradients $(q_j-q_i)$, e.g., $|q_j-q_i|^{2p-2}(q_j-q_i)$.

²

To simplify computations, we proceed with $2p$-Laplacians rather than $p$-Laplacians.

✖

The long-time behavior of the $p$-alignment model with $p>1$ is distinctly different from the pure alignment model when $p=1$. Specifically, Corollary 4.2 asserts a polynomial time decay of energy fluctuations when $p>1$, compared with exponential decay when $p=1$. These distinctly different time decay bounds are echoed throughout Section 5. In particular, it is the polynomial-in-time decay when $p>1$, which enables us to treat $p$-alignment with pressure in Section 6. We note in passing that there is yet a different behavior of finite time rendezvous for $p$-alignment when $0\leqslant p <1$, which we comment upon in Remark 5.5.

The large-crowd dynamics associated with 2.1 is captured by the corresponding hydrodynamic description