# 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 hydrodynamics, based on -alignment 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 -Laplacians*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 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. -alignment

## 1. Introduction—Alignment dynamics and entropic pressure

Alignment reflects steering toward an average heading Reference Rey1987. It plays an indispensable role in the process of emergence in swarming dynamics and, in particular—in flocking, herding, schooling, …, Reference VCBCS1995Reference CF2003Reference CKFL2005Reference CS2007aReference CS2007bReference Bal2008Reference Kar2008Reference VZ2012Reference MCEB2015Reference PT2017, as well as the formation of other self-organized clustering in human interactions and in dynamics of sensor-based networks Reference Kra2000Reference BeN2005Reference BHT2009Reference JJ2015Reference RDW2018Reference DTW2019Reference Alb2019; more can be found in Reference MT2014, §9, in the book series on active matter Reference BDT2017/19Reference BCT2022, and in the recent Gibbs’ Lecture Reference Tad2022a.

We discuss alignment dynamics in two parallel descriptions. Historically, alignment models were introduced in the context of agent-based description Reference Aok1982Reference Rey1987Reference VCBCS1995. In particular, our discussion is motivated by the celebrated Cucker–Smale model Reference CS2007aReference CS2007b, 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 Laplacians -graphReference HT2008Reference CFTV2010Reference HHK2010Reference Shv2021. 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., Reference CS2007aReference CDMBC2007Reference Bal2008Reference Ka2011Reference GWBL2012Reference JJ2015Reference LZTM2019Reference MLK2019Reference ST2020b. 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 Our goal is to make a systematic study of the long-time swarming behavior of hydrodynamic alignment, portrayed in Section -alignment.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 driven by -alignment,*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 driven by -alignment*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,

We further assume that the metric kernel is decreasing with the distance reflecting the typical observation that the intensity of alignment decreases with the distance. In particular, we address general metric kernels , whether decreasing or not, in terms of their *decreasing envelope* Observe that we do not place any restriction on the upper bound of . in particular, therefore, our discussion includes the important subclass of ;*singular* communication kernels Reference ST2017aReference DKRT2018Reference MMPZ2019Reference AC2021b.

### 1.2. Entropic pressure

System Equation 1.1 is not closed in the sense that the pressure is not specified—neither in terms of algebraic relations with nor do we specify the precise dynamics of , 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. .

### Why entropic pressure?

System Equation 1.1 falls under the general category of hyperbolic balance laws Reference Daf2016, Chapter III, and Equation 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, Equation 1.1a Equation 1.1a yieldsFootnote^{1}

Adding the entropic description of the pressure postulated in Equation 1.2 leads to the entropic statement for the total energy, ,

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

To further motivate this notion of entropic pressure, we appeal to its underlying *kinetic formulation*. The hydrodynamics Equation 1.1 corresponds to the large-crowd dynamics of agents with position/velocity governed by the celebrated agent-based alignment model of Cucker and Smale ,Reference CS2007aReference CS2007b,

The alignment dynamics is driven by a weighted graph Laplacian on the right of Equation 1.5*empirical distribution*

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

This passage from agent-based to macroscopic description is outlined in Appendix A.1. It was justified for smooth kernels Reference HT2008Reference CFTV2010Reference CCR2011Reference FK2019Reference NP2021Reference Shv2021 and at least mildly singular kernels Reference Pes2015Reference PS2019Reference MMPZ2019. In this context, the pressure or Reynolds stress tensor corresponds to the *second-order moments*

We observe that the kinetic description of pressure in Equation 1.6 is consistent with the entropic inequality postulated in Equation 1.2. Indeed, *internal energy* which quantifies microscopic fluctuations around the bulk velocity

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

with the so-called heat flux *equality* Equation 1.8. But here one encounters the familiar problem of lack of closure, which arises whenever one is dealing with the highest truncated *Maxwellian* induced by the thermal equilibrium of the system Reference Lev1996Reference Gol1998Reference Cer2003Reference Vil2003. In the current setup, however, the agent-based dynamics Equation 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* Equation 1.2.

### Scalar pressure

We discuss the case of scalar pressure law *assumes* a mono-kinetic closure,

which is realized in terms of zero pressure, *equality* in Equation 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 Equation 1.2 but otherwise require no algebraic closure. Indeed, our notion of entropic pressure becomes more transparent in the scalar case

Formal manipulation, Equation 1.10

We point out that the inequality Equation 1.11 is the *reversed* entropy inequality encountered for

### Entropic energy dissipation

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

which reflects the dissipativity of the total energy

One of the main aspects of this work is dealing with arbitrary pressure, without any specifics about the second-order closure for *The source of energy making the motion possible …are not relevant*” Reference VZ2012, §1.1. Here, we abandon a closure in the form of thermal equality Equation 1.8 and, instead, retain the inequality postulated in Equation 1.2, compatible with the dissipativity of internal fluctuations, which we argued for in Reference Tad2021, p. 501. In particular, our definition of a pressure in Equation 1.2 can be realized in any intermediate scale between the microscopic agent-based description, Equation 1.5, and the macroscopic hydrodynamics Equation 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

## 2. -alignment

We begin with the agent-based description,

The case ^{2} which is found in recent applications of neural networks Reference FZN2021, spectral clustering Reference BH2009, and semi-supervised learning Reference ST2019Reference Fu2021. In the context of alignment dynamics it was introduced in Reference HHK2010Reference CCH2014. We were motivated by the example of the Elo rating system Reference JJ2015Reference DTW2019, in which the alignment of scalar ratings

The long-time behavior of the *pure* alignment model when

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