Hydrodynamic Equations for Active Brownian Particles in the High Persistence Regime

TL;DR

This work derives a principled continuum description for active Brownian particles in the high-persistence regime by starting from a two-dimensional kinetic theory with an effective collision operator and applying the Chapman–Enskog expansion to obtain Navier–Stokes–like equations for density $\\rho$ and polarization $\\mathbf{q}$. The resulting hydrodynamic equations feature a density-dependent convection velocity $V_{\\text{eff}}(\\rho)$ and nonlinear diffusion terms whose coefficients are fixed by microscopic ABP dynamics, and they predict motility-induced phase separation (MIPS) as well as density–polarization coupling effects. Linear stability analysis reveals a density-driven instability above a critical density, while adiabatic elimination and 1D reductions expose regimes of damped density–polarization waves and coarsening dynamics; gravity introduces sedimentation and polarization alignment, enriching interfacial behavior. The framework exposes both the power and the limitations of kinetic-based hydrodynamics for active matter, highlighting the role of correlations and finite-persistence corrections for accurately capturing dense phase behavior and suggesting directions for extending the theory to more complex active systems and interfacial phenomena.

Abstract

In the high persistence regime of non-inertial active Brownian particles (ABP), polarization becomes a relevant dynamical field. Based on a recently proposed kinetic description for ABP, we derive Navier-Stokes-like equations for the density and polarization fields in this regime. Using the Chapman-Enskog method, all transport coefficients in the equations are obtained entirely in terms of the microscopic dynamics. A linear stability analysis of the homogeneous and isotropic state shows that the derived equations correctly describe the density instability associated to the motility induced phase separation. Numerical solutions of the equations in one spatial dimension show the need of an additional regularizing pressure term to saturate the system at high densities. With the inclusion of this term, the solutions illustrate in detail the clustering dynamics, with the formation of polarized regions at the interfaces, and the subsequent coarsening of domains, as well as particle accumulation in presence of gravity. Finally, the derived equations imply that, as an effect of the coupling with the polarization, damped density wave modes appear in the system which were verified with numerical simulations.

Figures (10)

• Figure 1: Collision schematic for ABP in the limit of infinite persistence. For simplicity we illustrate the case of $\mathbf{\hat{n}}_2=-\mathbf{\hat{n}}_1$. Particle directors are shown in red. The solid black lines represent the trajectories of the particles during the collision, while the dotted lines are the movement of the particles in the absence of it. The end points of these trajectories are $\mathbf{r}_i^{\rm{coll}}$ and $\mathbf{r}_i^{0}$, respectively. The effective displacements $\mathbf{\Delta}_i$ caused by the interaction are in blue.
• Figure 2: Eigenvalues of the coupled modes, $\rho$ and $q_\parallel$ as a function of the wavevector $k$ for $D_r=0.01$ and $\rho_0=0.61$ (left) and $\rho_0=0.67$ (right). The real (imaginary) parts are shown with solid blue (dashed orange) lines. Units have been chosen so that $V=\sigma=1$.
• Figure 3: Phase diagram in the $\rm{Pe}-\rho_0$ space. Above the dashed line, the brown region shows where the theory loses validity. The solid black line shows the critical density: above (below) the solid black line the unstable (stable) region is shown in orange (blue). Units were fixed by $V=\sigma=1$.
• Figure 4: Intermediate scattering function obtained in a simulation of $N=6400$ ABP placed in square box of size $L\approx178.9$ (average density $\rho=N/L^2=0.2$) for $k=2\pi/L$. The dots are the results of the simulations and the solid line is a fit to the function $F_\text{oscill}(t)=A e^{-t/\tau}\left[\cos\omega t+\sin\omega t/(\omega \tau) \right]$. Units have been chosen so that $V=\sigma=1$.
• Figure 5: Top: inverse values of the fitted relaxation times $\tau_{1,2}$ and $\tau$ as a function of the dimensionless wavenumber for simulations of $N=6000$ ABP in a rectangular box of lengths $L_x=3000$ and $L_y=10$ (average density $\rho=0.2$), and $\text{Pe}=100$. The dots are the results of the simulations and the solid line is the theoretical prediction. Bottom: fitted oscillation frequencies $\omega$. The dots are the results of the simulations, the solid line is the theoretical prediction, and the dashed line the fit to $\omega=C\sqrt{k^2-k_0^2}$, which gives $k_0=4.62\times 2\pi/L_x$. Units have been chosen so that $V=\sigma=1$.