Cage Breaking Far from Equilibrium

Abstract

Active matter can flow and yield under conditions where passive matter jams and slows down, as self-propulsion significantly modulates particle escape from local cages. How activity microscopically reshapes the caging environment to produce this effect, however, remains poorly understood. Here we study a minimal active-matter model of cage breaking: three distinguishable self-propelling disks under circular confinement. This simple setting allows us to construct an entropic landscape for rearrangements and to compare it exactly with its equilibrium counterpart. At low activity the landscape is effectively bistable, whereas at high activity it develops additional metastable basins associated with frustrated clusters at the boundary. We quantify the system's departure from equilibrium and show that cage breaking is fastest when the persistence length matches the particle radius, linking a geometric microscopic scale to the enhanced dynamics of active glasses. Extending the landscape to two dimensions reveals circulating probability currents, and a Markov-state description shows that detailed balance is broken both in the continuous landscape dynamics and in the coarse-grained transitions between entropic basins. Our results provide a minimal microscopic framework for understanding how activity reshapes caging, relaxation, and irreversibility in dense nonequilibrium matter.

Figures (12)

• Figure 1: Two-state model system for cage breaking becomes multistable with activity.A. Schematic of the mapping from bulk systems to our minimal model, and definitions of geometric parameters. B. Dynamics in $h$--space are intermittent for tight confinement ($\epsilon=0.01, \ell_p=0.2$), with the system spending long times between sign changes of $h$. C. Illustration of the effect of activity on the entropic landscape, with balanced transitions on a bistable landscape near equilibrium and unbalanced transitions on a multistable landscape far from equilibrium. D. Entropic landscape for the passive and active systems under tight confinement, $\epsilon=0.077$. The $\ell_p \to 0$ curve is the analytic result for a passive Brownian hard-disk system at this confinement (Ref. Hunter2012Free-energyDisks), while the $\ell_p=5$ curve is simulated.
• Figure 2: Persistent particles reshape their entropic landscape through clustering on the confinement boundary.A. Evolution of the entropic landscape as a function of the persistence time, $\epsilon=1$. Here we plot each landscape relative to its deepest minimum, and as a function of $|h|$ since the landscape is symmetric with respect to $h$. The landscape arising from sampling of stable configurations on the boundary is labeled $\ell_p=\infty$. Above: typical clustered configurations of the particles that cause the minima in the entropic landscape through activity frustration, with a black arrow showing their h values. Inset: the Wasserstein-1 distance $W_1$ between the simulated landscapes and the analytic distribution of the system at equilibrium (Ref. Hunter2012Free-energyDisks) monotonically increases with $\ell_p$ and quantifies the landscape's distance from equilibrium. Points are data from simulated landscapes, black curve is a fit to a sigmoid function. $W_{\mathrm{stable}}$ is the Wasserstein-1 distance of the stable configurations distribution at $\ell_p=\infty$. B. The locations of the system's entropic landscape minima as confinement strength $\epsilon$ varies, as predicted by geometry (solid curves) and extracted from stable configuration simulations with an extrema detection algorithm (open markers). C. Entropic depths of the points on the simulated landscapes from panel A at each minima location relative to the entropic barrier at $h=0$, as a function of persistence time. The crossover where the depth at $h_2$ becomes maximal indicates that it transitions to being the most sampled minima at high activity for this confinement size.
• Figure 3: Optimal cage breaking time at intermediate activity; caging dependence on confinement.A. The caging time is nonmonotonic with persistence length, minimized at an intermediate value $\ell_{p}^{\mathrm{min}}$ whose location does not depend on the self-propulsion force $f$. Points are simulated data, curves are spline fits to a quartic polynomial in log-log space. Here $\epsilon=0.062$ and $E_{\mathrm{int}}=1000$. Inset: Values of $\ell_{p}^{\mathrm{min}}$ extracted from fits of data with $f=0.025$, $E_{\mathrm{int}}=1000$ and various values of $\tau_p$ and $\epsilon$ is consistent with the particle's effective radius $r_{\mathrm{eff}}=1$ under tight confinement (small $\epsilon$), where cage breaking is well-defined with a single pathway. B. As the confinement size varies for fixed persistence length, the caging time finds an optimum at $\epsilon > 1$ (exact value depends on the simulated persistence length). Here and in panel C $E_{\mathrm{int}}=40$. Data are colored according to the legend in panel C. C. A modified version of Kramer's law (Eq. \ref{['eq: kramers-scaling']}) accurately captures caging times over all entropic barrier depths $S_b$ and levels of activity. The slope transitions with increasing $\ell_p$ from the value of 1 expected at equilibrium to smaller values as cage breaking is enhanced by activity. Caging time data is normalized by the prefactor $\tau_0(\ell_p)$ from a fit of Eq. (\ref{['eq: kramers-scaling']}) to each dataset.
• Figure 4: Markov state model for entropic basin transition rates and breaking of detailed balance.A. 2D entropic landscape, Markov state transition graph, and probability flux magnitude and streamlines map from near-equilibrium simulations $l_p=0.005$, $\epsilon=0.25$. Transition rates between states are symmetric near equilibrium, and probability fluxes are vanishing. B. Entropic landscape, Markov state transition graph, and probability flux plot for far-from equilibrium simulations $l_p=10$, $\epsilon=0.25$, showing highly asymmetric transition rates and substantial probability flux circulations. In both A and B, transition rates are represented on log scale by the widths of the arrows. Locations of the entropic landscape's local minima are labeled with larger circles than the threshold that was applied during graph computation for visibility.
• Figure 5: A. The distribution $n(|h|)$ for boundary-initialized stable configuration simulations, $\epsilon=1$. Theoretical peak locations at $h_{1}, \ldots, h_5$ are marked with vertical dashed lines. B. The corresponding entropic landscape, where the peaks in the distribution become local entropic minima. Note the lack of an entropic barrier at $h=0$, as by only sampling stable configurations we do not have any cage breaking transitions. Here we plot the values at the bin center locations of the $n(h)$ histograms, and slight apparent discrepancies between the simulated entropic minima locations and the theoretical predictions are due to finite-width bins.