Not-so-glass-like Caging and Fluctuations of an Active Matter Model

TL;DR

The paper addresses how activity alters dynamical arrest in dense active matter by studying a minimal active model, the ABP-RLG, across dimensions $d=2$ to $12$. It defines dimensionless controls $\widehat{\varphi}$ and $\widehat{\mathrm{Pe}}$ with $\mathrm{Pe}=v_0/\sqrt{D_t D_r}$ and $D_r = d D_t$, and analyzes the tracer's MSD and fluctuations to identify how activity changes caging. The key findings are that activity shifts the arrest density to higher values and saturates near the percolation threshold, while the tracer explores lower-dimensional cages and exhibits a distinct short-time peak in the non-Gaussian parameter that grows with $\widehat{\mathrm{Pe}}$. These results provide a concrete, dimension-dependent signature of active glassiness and highlight the need for DMFT with fluctuation corrections in first-principles descriptions.

Abstract

Simple active models of matter recapitulate complex biological phenomena. The out-of-equilibrium nature of these models, however, often makes them beyond the reach of first-principle descriptions. This limitation is particularly perplexing when attempting to distinguish between different glass-forming mechanisms. We here consider a minimal active system in various spatial dimensions to identify the processes underlying their sluggish dynamics. Activity is found to markedly impact cage escape processes and critical fluctuations associated with exploring lower-dimensional caging features.

Figures (4)

• Figure 1: Sample $d=2$ trajectory up to $t=30$ of a tracer with (a) $\widehat{\mathrm{Pe}} = 0$ (BP-RLG) and (b) $10$ (ABP-RLG) within the same fixed obstacles at $\widehat{\varphi} = 1.5$ (drawn to unit diameter). The passive tracer uniformly samples void space and rarely escapes the entropic bottleneck; the active tracer preferentially explores the surface of the obstacles and easily traverses the narrow path out. Activity alters the relaxation process. (c) Steady-state (bottom) radial distribution function of obstacles around the tracer, $g(r)$, and (top) corresponding effective potential for $\widehat{\mathrm{Pe}} = 0, 1, 10$ (blue to red lines). The contact peak markedly grows with activity up to several $k_\mathrm{B}T$, indicating a stronger effective attraction between the tracer and the obstacles.
• Figure 2: (a) MSD for $\widehat{\mathrm{Pe}} = 10$ for $\widehat{\varphi} = 0.5, 1, 2, 2.5$, and $4$ (from top to bottom), and $d = 3, 4, 6$, and $12$ (fading as $d$ decreases) compared with the free-space result from Eq. \ref{['eq:abp_msd_app']} (black dashed line). (Insets) For $d = 12$, (top) The ABP-RLG diffusivity at the same densities (lines from top to bottom) and (bottom) its ratio with the BP-RLG diffusivity. The long-time diffusivity of ABP-RLG is clearly enhanced. (b) The critical density $\widehat{\varphi}_\mathrm{d}$ for $d=12$ (squares) increases with $\widehat{\mathrm{Pe}}$ and appears to saturate around the percolation threshold $\widehat{\varphi}_\mathrm{p}(d=12)=2.64(9)$biroli2021interplay (black dashed line), and so does the scaling exponent $\gamma$ (circles). Results for $\widehat{\mathrm{Pe}}=0$ in the limit $d\rightarrow\infty$ -- $\widehat{\varphi}_\mathrm{d}=2.4034$ and $\gamma = 2.33786$ (empty symbols) -- are provided as reference biroli2021interplaykurchan2013exact. Lines are guides for the eye. (Inset) Critical scaling of the long-time diffusivity $\widehat{D}_\infty$ for $d = 12$ with $\widehat{\mathrm{Pe}}=1,2,3,5,10$ (blue to red lines).
• Figure 3: Time evolution of $\widehat{\alpha}_2$ for the ABP-RLG with (a) $\widehat{\mathrm{Pe}} = 5$ and various obstacle densities $\hat{\varphi}=1, 1.5, 2, 2.3, 2.38, 2.4,$ and $2.42$ (blue to red lines), and (b-g) $\widehat{\mathrm{Pe}} = 10$ in $d = 3, 4, 6, 8, 10,$ and $12$ (blue to red lines) for given obstacle densities. For $\widehat{\varphi}=0$, analytical expressions (dashed lines) can be obtained in all $d$SI. Note that results for $\widehat{\varphi}\gtrsim\widehat{\varphi}_\mathrm{p}$, in which cases the traces does not diffuse at long times, are faded out to clarify the dimensional trend, and the ranges corresponding to percolation physics are dotted out.
• Figure 4: Density evolution of the height of the first peak of $\hat{\alpha}_2$ for the ABP-RLG at $\widehat{\mathrm{Pe}}=10$ for various $d$ and obstacle densities. The peak height generally saturates as $d$ increases, but is roughly proportional to $d$ around $\hat{\varphi}_d$ similarly to the long-time peak of the BP-RLG charbonneau2024dynamics. (inset) The linear growth of the peak height with $\widehat{\mathrm{Pe}}$ for $d=12$ and $\hat{\varphi}=1$ (dashed line), $2.5$ (solid line) and $4$ (dot-dashed line).