Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Essentials of the kinetic theory of multi-agent systems

Nadia Loy, Andrea Tosin

TL;DR

This paper considers, in particular, scalar equations implementing linear symmetric interaction rules, for which the theory of well-posedness, trend to equilibrium, and Fokker-Planck asymptotics is developed by relying extensively on Fourier methods.

Abstract

In this paper, we present a critical collection of essential mathematical tools and techniques for the analysis of Boltzmann-type kinetic equations, which in recent years have established themselves as a flexible and powerful paradigm to model interacting multi-agent systems. We consider, in particular, scalar equations implementing linear symmetric interaction rules, for which we develop the theory of well-posedness, trend to equilibrium, and Fokker-Planck asymptotics by relying extensively on Fourier methods. We also outline the basics of Monte Carlo algorithms for the numerical solution of such equations. Finally, we elaborate the theory further for Boltzmann-type equations on graphs, a recent generalisation of the standard setting motivated by the modelling of networked multi-agent systems.

Essentials of the kinetic theory of multi-agent systems

TL;DR

This paper considers, in particular, scalar equations implementing linear symmetric interaction rules, for which the theory of well-posedness, trend to equilibrium, and Fokker-Planck asymptotics is developed by relying extensively on Fourier methods.

Abstract

In this paper, we present a critical collection of essential mathematical tools and techniques for the analysis of Boltzmann-type kinetic equations, which in recent years have established themselves as a flexible and powerful paradigm to model interacting multi-agent systems. We consider, in particular, scalar equations implementing linear symmetric interaction rules, for which we develop the theory of well-posedness, trend to equilibrium, and Fokker-Planck asymptotics by relying extensively on Fourier methods. We also outline the basics of Monte Carlo algorithms for the numerical solution of such equations. Finally, we elaborate the theory further for Boltzmann-type equations on graphs, a recent generalisation of the standard setting motivated by the modelling of networked multi-agent systems.

Paper Structure

This paper contains 39 sections, 27 theorems, 369 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Derivation of a homogeneous Boltzmann-type equation
  3. Agent-based model
  4. Statistical description
  5. Weak and strong forms of a Boltzmann-type equation
  6. Evolution of the moments
  7. The homogeneous Boltzmann equation
  8. Fourier transform in kinetic theory
  9. Fourier-transformed Boltzmann-type equation
  10. A priori regularity of the solution
  11. Fourier metrics
  12. Basic well-posedness theory
  13. Existence and uniqueness of the solution
  14. Continuous dependence on the initial datum
  15. Preservation of the support
  16. ...and 24 more sections

Key Result

Lemma 3.1

If $\mu\in\mathcal{P}(\mathbb{R})$ then $\hat{\mu}\in C^0_b(\mathbb{R})$ with $\Vert\hat{\mu}\Vert_\infty=1$.

Figures (2)

  • Figure 1: Numerical solution at successive computational times of the Boltzmann-type equation \ref{['eq:Boltztype.scaled']} in the quasi-invariant regime with interaction coefficients \ref{['eq:qeps']}, \ref{['eq:peps.eta']} and: left column $\epsilon=4\cdot 10^{-2}$, center column $\epsilon=10^{-2}$, right column $\epsilon=10^{-3}$.
  • Figure 2: Numerical solution at successive computational times of the Boltzmann-type equation \ref{['eq:Boltztype.scaled-advection']} in the advection-dominated quasi-invariant regime, cf. Section \ref{['sect:q.i._advection']}, and comparison with the exact solution \ref{['eq:g.FP-advection']} of the limit Fokker--Planck equation \ref{['eq:FP.strong-advection']} (solid line).

Theorems & Definitions (80)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 70 more