On reduced inertial PDE models for Cucker-Smale flocking dynamics

TL;DR

This work introduces a reduced inertial PDE for Cucker-Smale flocking that evolves only over space through the empirical density $\rho$ and momentum density $\boldsymbol{j}$. The authors derive 1D and multi-dimensional variants, including a space-weighted 1D model and a diffusive regularisation for $d>1$, and extend to a stochastic SPDE. They establish PDE flocking under a kernel split $a = C_a + \theta g$, provide an $H^{-2}$ error bound between the PDE and mollified particle dynamics in 1D, and demonstrate substantial computational savings and numerical fidelity compared to particle simulations and the hydrodynamic reduction. The framework directly links the reduced model to underlying particle dynamics, enables analysis of pre-flocking corrections via weights, and sets the stage for studying velocity fluctuations and control in flocking systems.

Abstract

In particle systems, flocking refers to the phenomenon where particles' individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behavior. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. In this paper, we introduce a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations.

Figures (8)

• Figure 1: Schematic overview of our approach for deriving the reduced PDE model from a particle model. Blue circles represent particles characterised by their position (center of the circle) and velocity (indicated by the arrows). In the flocking regime, particles align their velocities. PDE model is given by the empirical density $\rho$ and momentum density $\boldsymbol{j}$.
• Figure 2: Snapshots of the results obtained for the Cucker-Smale model \ref{['particles_det_x']}-\ref{['particles_det_v']} and reduced PDE model \ref{['reduced_CS_rho_d']}-\ref{['reduced_CS_j_d']} in two-dimensions at times $t=0$ (first column), $t=0.4$ (second column), and $t=2$ (third column). Top-row: particle system for $N=1000$, with the arrows and their base corresponding to the particles' velocities and positions respectively. Middle-row: the empirical density $\rho$ from the reduced model, brighter colours indicating a higher density. Bottom-row: the empirical momentum density represented as a vector field. The red arrow centred at the origin corresponds to $\overline{\boldsymbol{v}}$.
• Figure 3: Results obtained for the Cucker-Smale model \ref{['particles_det_x']}-\ref{['particles_det_v']}, reduced PDE \ref{['reduced_CS_rho']}-\ref{['reduced_CS_j']} and reduced PDE with the weight-dependent kinetic term \ref{['reduced_CS_rho_w']}-\ref{['reduced_CS_j_w']} in one dimension, where $w$ is given by \ref{['exp_weight']}. a) The empirical density $\rho$ and b) the empirical momentum density $j$, at $t=2$, evaluated for the particle system using a kernel with $\epsilon = 5$.
• Figure 4: The term $\mathbb{E} \left[ \left\lVert R(s) \right \rVert_{L^{2}}^2 \right]$, \ref{['eq:R_L2']} evaluated at $t=0$ for a variety of parameter settings. a) The dependence of \ref{['eq:R_L2']} on the initial velocity spread of the particle system for fixed $\epsilon = 0.1$. The velocity spread is adjusted by changing the width of the uniform distribution from which the initial velocities are drawn. Furthermore, the effect of the shape of the initial position distribution is also shown by varying $\kappa$ of the von-Mises distribution from which initial particles positions are drawn. b) Similar to a) but showing the dependence fot fixed velocity spread, and varying $\epsilon$.
• Figure 5: The discrepancy between the particle system and reduced PDE \ref{['eq:L2_err']} in one dimension, evaluated at $t=2$. a) The error for fixed $\epsilon$ and varying initial velocity spread with initial positions drawn from a uniform distribution centred at zero with width 40. The solid, black line is for the reduced PDE model without the inclusion of the weight \ref{['reduced_CS_rho']}-\ref{['reduced_CS_j']} and the dashed, blue line for the PDE \ref{['reduced_CS_rho_w']}-\ref{['reduced_CS_j_w']} with the inclusion of the weight \ref{['exp_weight']}. b) Similar to a), but for fixed initial velocity spread and varying $\epsilon$.

Theorems & Definitions (27)

• Definition 2.1: Particle flocking in $L^{p},L^{q}$
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5: PDE flocking
• Remark 2.6
• Theorem 3.1: PDE flocking: Informal statement combining Proposition \ref{['local_result']}, Proposition \ref{['local_result_w']}, Lemma \ref{['well_posedness_reg']}
• Theorem 3.2: Pathwise error between $\epsilon$-smoothed CS particle system and PDE system, $H^{-2}$ system
• Remark 3.3
• Remark 3.4