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Kinetic description and macroscopic limit of swarming dynamics with continuous leader-follower transitions

Emiliano Cristiani, Nadia Loy, Marta Menci, Andrea Tosin

Abstract

In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.

Kinetic description and macroscopic limit of swarming dynamics with continuous leader-follower transitions

Abstract

In this paper, we derive a kinetic description of swarming particle dynamics in an interacting multi-agent system featuring emerging leaders and followers. Agents are classically characterized by their position and velocity plus a continuous parameter quantifying their degree of leadership. The microscopic processes ruling the change of velocity and degree of leadership are independent, non-conservative and non-local in the physical space, so as to account for long-range interactions. Out of the kinetic description, we obtain then a macroscopic model under a hydrodynamic limit reminiscent of that used to tackle the hydrodynamics of weakly dissipative granular gases, thus relying in particular on a regime of small non-conservative and short-range interactions. Numerical simulations in one- and two-dimensional domains show that the limiting macroscopic model is consistent with the original particle dynamics and furthermore can reproduce classical emerging patterns typically observed in swarms.
Paper Structure (22 sections, 1 theorem, 114 equations, 7 figures, 2 tables)

This paper contains 22 sections, 1 theorem, 114 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Microscopic outlook
  3. Kinetic approach
  4. Stochastic particle model
  5. Boltzmann-type description
  6. Approximation in the regime of small non-conservative interactions
  7. Macroscopic limit
  8. The hydrodynamic limit
  9. On the local approximations of , in the hydrodynamic limit
  10. Numerical tests
  11. Numerical approximation
  12. Numerical grid
  13. Equation for
  14. Equation for
  15. Equation for
  16. ...and 7 more sections

Key Result

Theorem 4.4

Let $f=f(x,v,\lambda,t)$ be a distribution function such that: Then: and likewise

Figures (7)

  • Figure 1: Test 1D. Comparison between the numerical results of the macroscopic model \ref{['eq:macro.noncons']} with null Dirichlet boundary conditions (continuous lines) and of the microscopic stochastic process \ref{['def:micro_drift']}-\ref{['def:Sigma']} ('o' markers). A small numerical instability can be observed in the approximation of $\rho$ when the velocity changes sign and this is due to the upwind scheme used for the continuity equation.
  • Figure 2: Test 2Da: Initial conditions for (a) $\rho$ and (b) $l$.
  • Figure 3: Test 2Da: evolution of $\rho$ (first line) and $l$ (second line) at time a),d) $t=10$, b),e) $t=90$, c),f) $t=210$ . White arrows describe the velocity field.
  • Figure 4: Test 2Db: initial condition for $\rho$.
  • Figure 5: Test 2Db: evolution of $\rho$ (first line) and $l$ (second line) at time a),d) $t=20$, b),e) $t=40$, c),f) $t=180$ . White arrows describe the velocity field.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Remark 4.5