Structure and Memory Control Self-Diffusion in Active Matter

Abstract

Despite extensive progress in characterizing the emergent behavior of active matter, the microscopic origins of self-diffusion in interacting active systems remain poorly understood. Here, we develop a framework that quantitatively links self-diffusion to collisional forces and their temporal correlations in active fluids. We show that transport is governed by two contributions: an equal-time suppression of motion arising from anisotropic collisional forces, and a memory correction associated with the temporal persistence of these forces. Together, these effects yield an exact expression for the self-diffusivity in terms of measurable force statistics and correlation times. We apply this framework to purely repulsive active Brownian particles and find that self-diffusion is always reduced, with collisional memory acting as a strictly dissipative correction. Our results establish a direct connection between microscopic force correlations and macroscopic transport, providing a general mechanical perspective for interpreting self-diffusion in active matter.

Figures (6)

• Figure 1: Schematic of the active Brownian particle model (ABP) and force geometry. Particles self-propel along their orientation $\hat{\mathbf q}_i$ and interact via conservative forces acting along interparticle separation vectors $\mathbf r_{ji}$. The projection of the collisional force onto the propulsion direction, $F_{\parallel}\equiv \mathbf F_c\cdot \hat{\mathbf q}$, generates an anisotropic force imbalance that suppresses particle motion and underlies the reduction of self-diffusion.
• Figure 2: Simulation results for the 2D ABP system. (a) Parameter space explored as a function of Péclet number $\mathrm{Pe}$ and area fraction $\phi$. The solid black line denotes the binodal separating homogeneous states from the two-phase coexistence region for hard active Brownian disks Levis2017-je. Light and dark gray regions indicate the hexatic and solid phases for passive hard disks, respectively lee2016molecular. The red dotted line marks close packing in 2D, $\phi_{cp} \approx 0.91$. State points enclosed by the black oval exhibit anomalous diffusion at intermediate times. (b--i) Mean-squared displacement (MSD) for states at the indicated values of $\mathrm{Pe}$ and $\phi$. The MSD and lag time are rescaled by $\langle \mathbf v^2\rangle \tau_R^2$ and $\tau_R$, respectively, collapsing the short-time ballistic regime. The solid black line shows the MSD of an ideal 2D ABP.
• Figure 3: Normalized long-time self-diffusivity $D/D_0$ as a function of area fraction $\phi$ for various Péclet numbers. Across the parameter range studied, increasing $\phi$ leads to a monotonic suppression of self-diffusion. The solid black line shows the analytical prediction for $D/D_0$ in the asymptotic limit $\mathrm{Pe}\to\infty$Soto2025-lg. The dotted line corresponds to the analytical expression for passive Brownian disks Stopper2018-nk.
• Figure 4: Structural origin of the collisional force variance. (a) Collisional force variance $\langle \mathbf F_c^2\rangle$ as a function of area fraction $\phi$ for various Péclet numbers. For $\mathrm{Pe}>1$, $\langle \mathbf F_c^2\rangle$ grows approximately linearly with $\phi$; the solid black line corresponds to $\langle \mathbf F_c^2\rangle/(\gamma U_a)^2=\phi$. (b) Parity plot comparing $\langle \mathbf F_c^2\rangle$ measured directly from simulations to the theoretical prediction based on the anisotropic pair correlation function [Eq. (\ref{['eq:16']})]. The collapse onto the parity line demonstrates that the collisional force variance is quantitatively determined by local pair structure with no adjustable parameters. (c) Force-weighted dipolar anisotropy $\mathcal{G}$ as a function of $\phi$ for various Péclet numbers. For $\mathrm{Pe}>1$, $\mathcal{G}$ depends only weakly on $\phi$, indicating that increasing $\langle \mathbf F_c^2\rangle$ is primarily driven by the number of near-contact neighbors rather than changes in angular structure.
• Figure 5: Collisional correlation timescales modulating the self-diffusivity. (a) Collisional force-orientation correlation time $\tau_{ca}$. (b) Collisional force autocorrelation time $\tau_{cc}$. (c) Net collisional memory timescale $\tau_D = \tau_{ca} - \tau_{cc}$, which quantifies the contribution of collisional memory to long-time transport.