Abstract
In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a follow-the-leader interaction rule. Due to the Hilbert transform, the swarm dynamic is described by nonlinear and non-local dynamics that can be solved exactly. From the mutual interactions emerges a fat-tail soliton that can be obtained in a closed analytic form. The soliton median evolves nonlinearly with time. This behaviour can be clearly understood from the interaction of mutual agents.
Highlights
The dynamic of a Markov process Xt ∈ R is fully characterised by its transition probability density (TPD) P(z, t|y, 0), solving a differential Chapman–Kolmogorov (DCK) [1]:
Let us emphasise that it is truly remarkable that the nonlinearity together with the nonlocal character (Hilbert transform) of the swarm dynamics still allow exact results to be obtained in the short-range interaction regime
Besides the fundamental relevance of the BO in fluid dynamics, it is remarkable that adding an extra dissipative term enables us to describe the collective motion of swarms of Cauchy processes
Summary
The dynamic of a Markov process Xt ∈ R is fully characterised by its transition probability density (TPD) P(z, t|y, 0), solving a differential Chapman–Kolmogorov (DCK) [1]:. Besides the fluid domain, where Equations (3) and (8) play a central role, these enter into the realm of mean-field (MF) multi-agent (swarms) dynamics This is the aspect that will be discussed here. Our paper explores the swarm dynamics of agents driven by Cauchy processes. With such jump noise sources and a follow-the-leader type algorithm, we will show the emergence of fat-tail solitons at the macroscopic level. Everything in this program can be worked out exactly, and specific technical details are listed in a couple of appendices. Note that for finite dimensional systems, a similar type of regulator was implemented for diffusive dynamics in [20]
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