Critical Assessment of the Boltzmann Approach to Active Systems

Generic models of propelled particle systems posit that the emergence of polar order is driven by the competition between local alignment and noise. Although this notion has been confirmed employing the Boltzmann equation, the range of applicability of this equation remains elusive. We introduce a broad class of mesoscopic collision rules and analyze the prerequisites for the emergence of polar order in the framework of kinetic theory. Our findings suggest that a Boltzmann approach is appropriate for weakly aligning systems but is incompatible with experiments on cluster forming systems.

Figure 1

(a) Rod interactions within an angular alignment range $\overline{\psi }$ (grey shaded) lead to polar alignment, whose direction typically deviates from the half-angle alignment $w=\frac{1}{2}$ [$w$ is directly given by Eq. (1)]. (b) Simulated example trajectory for a model of propelled rigid rods (relative angle $\psi \approx \pi /3$, impact parameter $b\approx 0$ in units of rod diameter, $w(\pi /3,0)\approx 1.3$).

Figure 2

Binary scattering plot: relative postcollision angle ${\psi }^{\prime }={\theta }_1^{\prime }-{\theta }_2^{\prime }$ (scale bar), versus impact parameter $b$ and relative precollision angle $\psi$, for propelled hard rods (a), and stiff polymers (b). The mesoscopic alignment weight $w(\psi )$ and the alignment range $\overline{\psi }$ are indicated as crosses and vertical dotted lines, respectively. $L=10$; for simulation details and videos refer to Supplemental Material [22].

Figure 3

(a) Transition density ${\rho }_t(\overline{w})$: curves from left to right (dash-dotted, dashed, solid) correspond to $\sigma =\{0.7,0.5,0\}$ with $\overline{\psi }=\pi$. (b) ${\rho }_t^{-1}(\overline{w},\overline{\psi })$ for $\sigma =0.5$. Parameters for (a) and (b): $L/d=1$, ${\sigma }_0=0.5$. Stationary numerical solution of Eq. (2): (c) Snapshots of observed polar patterns (periodic boundary conditions; arrows: normalized, local polarization) for $\overline{\psi }=3\pi /4$: homogeneously distributed density field (top, $\overline{w}=0.8$) and polar-wave pattern (bottom, $\overline{w}=0.92$, relative density differences are indicated, see color bar). (d) Total polarity $\mathcal{P}$ (solid lines) and spatial density variance $\mathcal{D}$ (dashed lines) versus $\overline{w}$ (from left to right: grey-green-red-blue correspond to $\overline{\psi }=\{\pi ,3\pi /4,\pi /2,\pi /4\}$). For $\psi =\pi /4$ (blue), $\mathcal{D}=0$ and $\mathcal{P}=0$ for all $\overline{w}$. Parameters for (c) and (d): ${\rho }_0=0.15$, $\sigma ={\sigma }_0=0.15$, $L/d=1$.