Non-reciprocal anti-aligning active mixtures: deriving the exact Boltzmann collision operator

TL;DR

This work addresses binary mixtures of self-propelled particles with non-reciprocal anti-aligning interactions by deriving an exact active Boltzmann equation with a collision operator tailored to one-sided molecular chaos. The authors account for phase-space compression and correlation buildup, obtaining an expression for the collision term that includes unique non-reciprocal contributions via $Sc^+$ and $Sc^-$, and they present a perturbative expansion valid at low density and small coupling. Their results extend previous reciprocal-focused theories and show that non-reciprocity shifts the flocking transition and endows the system with density- and velocity-dependent ordering, including predictions for the self-diffusion coefficient. Comparisons with agent-based simulations demonstrate quantitative agreement for time evolution, stationary states, and transport properties within the dilute, weak-coupling regime, validating the Boltzmann framework beyond mean-field. The findings highlight the utility of exact collision operators for understanding non-reciprocal active matter and pave the way for exploring spatial patterns and higher-density regimes.

Abstract

We consider the effect of non-reciprocity in a binary mixture of self-propelled particles with anti-aligning interactions, where a particle of type A reacts differently to a particle of type B than vice versa. Starting from a well-known microscopic Langevin-model for the particles, setting up the corresponding exact N-particle Fokker-Planck equation and making Boltzmann's assumptions of low density and one-sided molecular chaos, the non-linear active Boltzmann equation with the exact collision operator is derived. In this derivation, the effect of phase-space compression and the build-up of pair-correlations during binary interactions is explicitly taken into account, leading to a theoretical description beyond mean-field. This extends previous results for reciprocal interactions, where it was found that orientational order can emerge in a system with purely anti-aligning interactions. Although the equations of motion are more complex than in the reciprocal system, the theory still leads to analytical expressions and predictions. Comparisons with agent-based simulations show excellent quantitative agreement of the dynamic and static behavior in the low density and/or small coupling limit.

Figures (5)

• Figure 1: Time evolution of the variance of the sum of all directional angles $\langle S^2\rangle$. For reciprocal interactions (blue dashed) and zero noise, $S$ is a conserved quantity. In the case of non-reciprocal interactions (green), it fluctuates strongly. Non-reciprocity is introduced by having two different species of particles (1 and 2), where the coupling constant $\Gamma_{\sigma_i,\sigma_j}$ between particle $i$ and $j$ now depends on the species $\sigma_i,\sigma_j\in{1,2}$ of the interacting particles. Simulation parameters are $Sc(1,1)=Sc(2,2)=0.02$, furthermore $Sc(1,2)=Sc(2,1) = 0.015$ in the reciprocal case, $Sc(1,2) = 0.015$, $Sc(2,1) = 0.005$ in the non-reciprocal case. The coupling strength is defined as $Sc(\alpha,\beta)=-\frac{\Gamma_{\alpha,\beta}R}{v_0}$. The particle number is $N_1=N_2=500$, the scaled density $M_1=M_2=0.03$ and $R=v_0=1$. The integration timestep is $\Delta t = 0.02$. The agent based simulations are compared to an analytical expression (orange) derived further below, see Eq. \ref{['eq:deltaSsqu']}. Subfigure (a) shows the situation without external noise. In (b) there is some external noise $\sqrt{2}\nu=10^{-3}$. As $\nu\ll \min(1,Sc)$, this noise has no influence on a collision of two particles. In this situation the non-reciprocity causes an additional change in $S$ that is independent from the stochastic noise term.
• Figure 2: Time evolution of the first two Fourier modes of the two species obtained from the theory (red and blue solid lines) in comparison with agent-based simulations (black dashed lines). Species 1 is shown on the left, species 2 on the right. Parameters are $M_1=0.03$, $M_2=0.06$ and $Sc(1,1)=0.01$, $Sc(2,2)=0.018$, $Sc(1,2)=0.01$, $Sc(2,1)=0.02$. Here, and in the following figures, no external noise was included, $\nu=0$. The difference between simulation and theory when the modes are very close to zero is an artifact of the numerical evaluation of the simulation, where it was more efficient to only measure the absolute value of the modes.
• Figure 3: (a) and (b) show where the disordered solution of the Boltzmann equation (blue) and the mean field equation (gray+blue) becomes linearly unstable. Figure (a) illustrates the influence of non-reciprocity on the location of the transition. All other parameters are symmetric between the two species in the left frame ($Sc(1,1)=Sc(2,2)=0.01$, $M_1=M_2=0.03$). In the right frame the species also differ in intra-species-coupling strength and density, ($2Sc(1,1)=Sc(2,2)=0.01$, $4M_1=M_2=0.03$). Figure (b) illustrates the influence of the density ratio $M_1/M_2$ on the location of the transition. In the left frame of (b), the interactions are reciprocal ($2Sc(1,1)=Sc(2,2)=0.01$, $Sc^-(1,2)=0$), in the right frame they are non-reciprocal ($2Sc(1,1)=Sc(2,2)=0.01$, $Sc^-(1,2)=-0.016$). Along the short black lines, agent-based simulations were performed for comparison. The red dot shows where the transition to an ordered stationary state takes place in the agent-based simulation. (c) and (d) show the group of simulations marked in (a) and (b). Blue dots are the stationary order parameters $|\hat{p}_1(1)|$ measured in the simulation, whereas red triangles represent the stationary values predicted by the theory. Grey crosses show the variance of the order parameter measured in simulation. The maximum of this variance was used to determine the exact location of the transition. The purple dotted line is the mean field stability line. The simulations were performed along lines with constant $Sc^-$ ($Sc^-=-0.00280435$ in (c) and $Sc^-=-0.016$ in (d)).
• Figure 4: Velocity autocorrelation of species 1 (blue) and species 2 (red). Solid colored lines refer to agent-based simulations, while the dashed black lines show the predictions from Eq. \ref{['eq:nonrec:tau']}. Parameters are $M_1=M_2=0.03$ and $Sc(1,1)=Sc(2,2)=0.01$, $Sc(1,2)=0.005$, $Sc(2,1)=0.002$.
• Figure 5: A version of Fig. \ref{['fig:nonrec:overview']}(c) with the same relative values of coupling and density, but higher absolute values: $Sc(1,1)=Sc(2,2)=0.1$, $Sc^-=-0.0280435$, and $M_1=M_2=15.62$. Blue dots are the stationary order parameters $|\hat{p}_1(1)|$ measured in the simulation, whereas red triangles represent the stationary values predicted by the theory. Grey crosses show the variance of the order parameter measured in simulation. The maximum of this variance was used to determine the exact location of the transition. The purple dotted line is the mean field stability line.