On a macroscopic limit of a kinetic model of alignment

TL;DR

This paper investigates the macroscopic behavior of a one-dimensional kinetic model of alignment, deriving diffusion and traveling wave equations as limits near different equilibrium states, depending on the sensitivity parameter.

Contribution

It provides a rigorous derivation of macroscopic limits from a kinetic alignment model, including diffusion and traveling wave equations, for different equilibrium regimes.

Findings

Diffusive limit yields a classical linear diffusion equation.

Aligned state leads to traveling wave solutions.

Results connect microscopic interactions to macroscopic PDEs.

Abstract

In the present paper the macroscopic limits of the kinetic model for inter-acting entities (individuals, organisms, cells) are studied. The kinetic model is one-dimensional and entities are characterized by their position and orientation (+/-) with swarming interaction controlled by the sensitivity parameter. The macroscopic limits of the model are considered for solutions close either to the diffusive (isotropic) or to the aligned (swarming) equilibrium states for various sensitivity parameters. In the former case the classical linear difusion equation results whereas in the latter a traveling wave solution does both in the zeroth (`Euler') and frst (`Navier-Stokes') order of approximation.