A microscopically reversible kinetic theory of flocking

TL;DR

This work develops a microscopically reversible kinetic theory for a two-species hard-sphere system (birds and air) in which a reactive collision $\mathrm{B}+\mathrm{A}+C_+ \leftrightarrow \mathrm{B}+\mathrm{A}+C_- $ transfers chemical energy into kinetic energy with energy change $\Delta E$, and introduces a chemostat to drive the system out of equilibrium via affinity $\Delta \mu$. Through coarse-graining, a momentum-source term $\mathbf S$ is extracted and identified with an $\alpha$-coefficient in the Toner–Tu equations, with $\mathbf S = -\alpha\mathbf u_0 + O(u_0^3)$ and $\mathbf u_0 = \mathbf v_0 - \mathbf w_0$; under grazing interspecies collisions and sufficient activity, $\alpha$ can become negative, signaling a flocking transition. In the passive limit, an $H$-theorem ensures relaxation to Maxwell–Boltzmann equilibria with $\rho^{+} = \rho^{-} e^{-\beta_A\Delta\mu}$, while active driving and grazing selectivity enable a velocity-growth regime that mirrors flocking. The framework thus connects microscopic, reversible reactions to emergent continuum flocking behavior and suggests pathways to full hydrodynamic descriptions and molecular-dynamics validation.

Abstract

We formulate a kinetic theory of two species of hard spheres which can undergo collisions converting chemical energy into kinetic energy. As the two species represent birds and air, this reactive collision mimics birds flapping their wings, allowing for their propulsion. We demand microscopic reversibility of the reactive collisions. We then introduce a chemostat to drive the system out of equilibrium. When the chemostat is sufficiently turned on and one restricts to grazing interspecies collisions, the momentum damping term can turn into a momentum growth term, hinting at a flocking transition.

Figures (6)

• Figure 1: Depiction of a binary hard sphere collision between two bird particles with ingoing and outgoing relative velocity $\mathbf g$ and $\mathbf{g}'$ as well as the scattering angle $\chi$ and impact parameter $b$.
• Figure 2: 3D plot of $G^+ ( u, \chi )$ within the $\chi$-domain specified by \ref{['eq:domain']}. We took $a=3$.
• Figure 3: Two-dimensional depiction of (a) a forward reactive bird-air collision and (b) an inverse reactive bird-air collision when the collision takes place inside the "reactive volume" $\kappa^{\pm} (u , \chi )$. The ingoing and outgoing relative velocities are drawn from the rest of the bird before and after the collision respectively. Note that it holds for any forward or inverse reactive collision that $\psi^{ + } \geq \psi^{ \prime + }$ and $\psi^{ - } \leq \psi^{ \prime - }$.
• Figure 4: $\alpha$ as a function of $\Delta \mu$, $\chi_{\text{graze}}$ and ${k_\text{b}} T_{{\text{A}} }$ . If not varied, we took $\Delta \mu = 5$, $\chi_{\text{graze}} = 0.2$, ${k_\text{b}} T_{{\text{A}}} = 2$, $a = 3$, $K=1$ and $m_{{\text{A}}} = m_{{\text{B}}} = 1$.
• Figure 5: Diagrams showing the sign of $\alpha$ for different values of $\Delta \mu$, ${k_\text{b}} T_{\text{A}}$ and $\chi_{\text{graze}}$. If not varied, the parameters are the same as those in Fig. \ref{['fig:three-subfigures']}.