Spatial and Temporal Cluster Tomography of Active Matter

Abstract

Critical phase transitions have proven to be a powerful concept to capture the phenomenology of many systems, including deeply non-equilibrium ones like living systems. The study of these phase transitions has overwhelmingly relied on two-point correlation functions. In this Letter, we show that cluster tomography -- the study of one-dimensional cross-sections of the clusters that emerge near a phase transition -- is an alternative higher-order tool that efficiently locates and characterizes phase transitions in active systems. First, using motility-induced phase separation as a paradigmatic example, we show how complex geometric features of clusters, captured by spatial cluster tomography, can be used to measure critical exponents in active systems without explicitly introducing system-specific order parameters. Second, we introduce temporal cluster tomography, an analogous cluster-based measurement that characterizes the dynamical behavior of active systems. We show that cluster dynamics can be captured by a generalization of burstiness analysis in complex temporal networks. Both spatial and temporal cluster tomography are easy to implement yet powerful approaches to study non-equilibrium systems, making them useful additions to the standard toolbox of statistical physics.

Figures (4)

• Figure 1: Cluster tomography.(a) Spatial cluster tomography. Given a cluster-forming system, we consider one-dimensional cross-sections and tally the distance $s$ between successive observations of each cluster, to detect and characterize phase transitions. (b) Temporal cluster tomography. Given a pair of particles (red), we measure the time intervals $s_t$ between successive observations of them belonging to the same cluster (first and third snapshots) to study the dynamics of the system.
• Figure 2: Spatial cluster tomography.(a) Gap-size statistics at $\mathrm{Pe}_r=10$ and volume fractions below, close to, and above the critical region. (b) Fraction of the available volume occupied by the largest cluster. (c) Corner contribution $\mathcal{C}$, excluding the largest cluster. The dashed line marks $\phi^* = 0.515$. (Inset) Data collapse of the corner contribution using the Ising values $b=0.0608$, $\nu=1$. $L \leq 32$ data are omitted for clarity. (d) Finite-size estimates of the lacunarity using $\ell=L/8$. An infinite-size extrapolation in $\ln\ell/\ell$ shows consistency with the Ising lacunarity (dashed horizontal line). Where no error bars are visible, the uncertainty is smaller than the marker.
• Figure 3: Temporal cluster tomography.(a) Representative time series for a particle pair at volume fractions below, at, and above the critical region over 200 time steps ($L=512$). Vertical lines mark snapshots where the pair is in the same cluster. (b) Temporal gap-size statistics at the same volume fractions as (a), normalized by the number of snapshots $T$. (c) Burstiness parameter calculated from $g_t(s_t)$. To avoid artefacts due to the periodic boundaries, only temporal gaps $s_t<L$ are used in the calculation. The dashed line marks $\phi^*=0.515$. Where no error bars are visible, the uncertainty is smaller than the marker.
• Figure 4: First-order transition. Gap-size statistics (a), corner contribution (b), temporal gap-size statistics (c), and burstiness parameter (d) across the MIPS boundary at $\mathrm{Pe}_r=100$, $L=256$.