Contact temporal network during motility induced phase separation

TL;DR

This work develops a temporal-network framework to characterize Motility-Induced Phase Separation (MIPS) in Active Brownian Particles (ABPs). By mapping ABP contacts within finite time windows to adjacency matrices, the authors analyze degree distributions $P(k)$, global clustering $C$, and average path length $l_G$ across single-phase and phase-separated regimes. In the single-phase regime, $P(k)$ is Gaussian and exhibits universal scaling with $\mu \propto \text{Pe} W$ and $\sigma_{st} \propto \sqrt{\text{Pe} W}$, leading to a collapse $P(k_{\text{peak}})=\Lambda/\sqrt{\mu}$; in MIPS, the dilute phase shows a stable peak near $k\sim20$, while the dense phase develops a hexagonal-caging signature at short windows and a broad plateau up to $k\sim100$ for longer windows. Global clustering and path-length analyses reveal phase-specific topologies: dilutes exhibit decreasing $C$ while dense phases maintain higher $C$ early on, and high mobility states show small $l_G$ with strong time-window collapse. Overall, the temporal-network approach exposes universal and phase-specific structures in active matter and provides a generalizable tool for probing non-equilibrium phase behavior and clustering dynamics.

Abstract

Motility-induced phase separation (MIPS) is a paradigmatic non-equilibrium transition in active matter, determined by the Péclet number and packing fraction. We investigate the single-phase and phase-separated regimes of MIPS using a complex network approach, where networks are constructed from particle interactions over finite time windows. In the single-phase (gas-like) regime, the degree distributions $P(k)$ exhibit Gaussian behavior and resemble those of random graphs. Plotting the location and height of the $P(k)$ peak reveals a universal curve across different Péclet numbers at fixed packing fraction. In the phase-separated regime, we analyze the dense and dilute phases independently. The $P(k)$ distributions unveil distinct collective dynamics, including caging in the dense phase and the emergence of active solid-like structures at longer times. Clustering coefficients and average path lengths in both phases stabilize rapidly, indicating that short simulations are sufficient to capture essential network features. Overall, our results show that network metrics expose both universal and phase-specific aspects of active matter dynamics. Notably, we identify distinct and previously unreported topological structures arising in the dense and dilute phases within the MIPS regime.

Figures (6)

• Figure 1: Snapshots of the system containing $N = 6 \times 10^{3}$ active Brownian particles. Each particle is colored according to the number of neighbors within a cut-off distance $C_{r}$. (Left) Dilute regime with packing fraction $\phi = 0.20$ and low Péclet number, $\text{Pe} = 10$. (Right) Dense regime with $\phi = 0.48$ and high Péclet number, $\text{Pe} = 100$, where motility-induced phase separation (MIPS) is visible.
• Figure 2: Schematic of particle encounters over a discrete time window. Panels (a--c) show particle positions at three consecutive time steps, with circles of radius $C_r$ indicating each particle’s interaction region. As particles move, reciprocal links form whenever two particles fall within each other’s interaction radius. Panel (d) shows the resulting adjacency matrix $A$ and its graph representation, where $A_{ij}=1$ if particles $i$ and $j$ interacted at any time in $W=\{t_1,t_2,t_3\}$, and $A_{ij}=0$ otherwise.
• Figure 3: Top: Degree distributions of systems in the single-phase regime for $\phi=0.2$ and different Péclet numbers: $\text{Pe} = 10$ and $\text{Pe} = 120$ for panels (a) and (b), respectively. The degree distributions are truncated when $P(k)=0$. Color indicates the width of the time window considered, transitioning from darker blue (shorter window) to darker red (longer window) as the window increases. Panel (c) shows the degree distribution of a gas of passive Brownian particles with a higher diffusion coefficient. Similarly, color indicates time window width, ranging from dark blue (short) to dark red (long). Bottom: Panel (d) shows the maximum value of $P(k_{\text{peak}}) \sim k_{\text{peak}}^{-\gamma}$ that decays as a power law with a critical exponent $\gamma = 0.5$, for different Péclet number $\text{Pe} \in [10, 120]$ at different time windows, including the passive Brownian gas. Panel (e) and (f) shows the relation followed by $\mu(\mathrm{Pe},W)=0.18\;\mathrm{Pe}\;W_i$ and $\sigma_{\mathrm{st}}(\mathrm{Pe},W)=\sqrt{0.18\;\mathrm{Pe}\;W_i/A}$ with $A=2\pi \Lambda^2$, in terms of the time windows $W_i=D_rt$ and $\mathrm{Pe}$ respectively.
• Figure 4: Log-log plots of the time‐averaged degree distribution $P(k)$ for four observation windows $W_i$ during motility-induced phase separation at $\phi = 0.48$ and $\text{Pe}=100$. (a) Dilute phase: $P(k)$ is non-Gaussian, with the highest probability at $k\sim20$ and an abrupt drop once $k\!\approx\!100$, the value corresponding to a particle that connects to a full row of neighbours across the box. The qualitative shape is unchanged by the choice of $W_i$, but the amplitudes differ slightly. (b) Dense phase: At early times (blue), the distribution rises sharply at $k=6$, then flattens into a plateau for $6 \lesssim k \lesssim 100$. The plateau becomes more pronounced for larger windows (green). (c) Semi-log average degree distribution during the dense phase, for three time windows. The plateau has two equally probably limits: one near $k \approx 10$, associated with particle caging in quasi-hexagonal structures (red square), and another near $k \approx 100$, linked to long-range connections within active solid regions (red circle). (d) Particle trajectories in the dense phase for a time window $W = 13.5t_r$. Particles with a small number of connections (red square in (c)) are shown in red, while those with a large number of connections (purple circle in (c)) are shown in purple. Gray particles represent a static image of the formed cluster.
• Figure 5: Global clustering coefficient ($C$) and total number of triplets and triangles $\#$ for the two-phase and single-phase regions across different time windows, respectively. (a)-(b) System during MIPS: Blue triangles represent the dilute region, green dots correspond to the dense region. (c)-(d) One phase case: The global clustering coefficient $C$ and the number of triangles and triplets $\#/$ for different Péclet numbers $\text{Pe}$.