Non-equilibrium thermodynamics in driven macroscopic self-assembly

TL;DR

The paper addresses non-equilibrium phase-space dynamics in self-assembly by using a macroscopic capillary-bound particle system driven with a Faraday-wave bath, enabling complete observation of microstate transitions and direct measurement of entropy production. It combines experiments with a many-body active Ornstein–Uhlenbeck model to capture correlated active forces and memory effects, achieving quantitative agreement on transition statistics and the nonzero entropy-production rate $\sigma$ that signals broken detailed balance. The study reveals persistent cyclic currents in state space (e.g., $\text{T} \rightarrow \text{C} \rightarrow \text{P}$) that bias dynamics away from equilibrium, and extends the framework to seven-particle clusters with a richer topology, maintaining nonequilibrium fluxes. By linking macroscopic active matter to microscopic active systems, the work offers a versatile platform for programmable non-equilibrium self-assembly and insights applicable to active colloids and bacterial baths.

Abstract

Equilibrium statistical mechanics provides a robust framework for characterizing phase transitions in systems whose microsopic dynamics are time-reversible. Efforts to develop and validate theoretical frameworks for time-irreversible, non-equilibrium systems are constrained by experimental data that capture only partial measurements of the system dynamics. We herein overcome this limitation using a tunable macroscopic platform for non-equilibrium physics: millimetric spheres bound by capillary attractions at the fluid interface and driven out of equilibrium by a field of supercritical Faraday waves. The external driving induces correlated fluctuations in the particle trajectories, which in turn excite structural rearrangements between distinct metastable cluster topologies. By tracking all microstate transitions experimentally, we directly measure a non-zero entropy production rate reflecting broken detailed balance and quantifying the system's departure from equilibrium. The measured stochastic dynamics are in quantitative agreement with a many-body active Ornstein-Uhlenbeck model, thus establishing a bridge to a wider class of athermal, self-propelled systems at the microscale. These results invite parallel studies of non-equilibrium self-assembly kinetics using active colloids or passive particles immersed in bacterial baths whose dynamics and irreversibility are likewise governed by correlated active forces and tunable inter-particle interactions.

Figures (4)

• Figure 1: Oblique perspective of a cluster of six spherical particles, each of radius 0.079 centimeters, momentarily occupying a triangle configuration in a field of supercritical Faraday waves of characteristic wavelength $\lambda = 0.84$ centimeters. The Faraday waves are visualized using a reflected color pattern harris2017visualization. The complex fluid environment due to the Faraday waves and horizontal turbulent surface flows plays the role of an active bath.
• Figure 2: Faraday-wave driven structural rearrangements of capillary-bound six-particle clusters. (a) Oblique images of the unique ground-state configurations formed when fewer than six spherical particles of radius 0.079 centimeters (cm) are placed on the quiescent surface of a water-glycerol mixture. (b) Addition of a sixth particle forms one of three metastable configurations: triangle (T), chevron (C, ground state), or parallelogram (P). (c) Schematic of our experimental set-up. Four clusters are confined to square-shaped corrals filled with a water-glycerol mixture. The entire set-up is vibrated vertically with acceleration $\Gamma$ and frequency $f$ above the Faraday threshold $\Gamma_F = 0.9g$. Typical experimental cluster trajectories (solid lines) over a 15 second interval are shown for (d) $\Gamma = 1.0g$ and (e) $\Gamma = 1.2g$. The white-to-red colormap along the cluster trajectory history indicates increasing time. The inset of both (d) and (e) plots the long-time experimental distribution of $v_c = \lVert\sum_{i = 1}^6 \dot{\mathbf{x}}_i(t)\rVert_2$, the instantaneous speed of the cluster center of mass. The mode of the distribution is 0.83 cm/s for $\Gamma = 1.0g$ and 1.16 cm/s for $\Gamma = 1.2g$ showing a speed enhancement of the particles as $\Gamma$ is increased. (f) Example of a P-to-C transition when a single bond between two particles (shaded in pink) breaks. A typical transition occurs over a sub-second timescale (0.18 s in this example).
• Figure 3: Control of six-particle assembly and transition statistics with vibrational forcing. Transition probability matrices for (a)--(b) $\Gamma = 1g$ and (c)--(d) $\Gamma = 1.2g$. The top figure in each pair are the experimental results, the bottom figure the corresponding theoretical prediction. The colors in (a)--(d) correspond to the conditional probability Pr(end$\lvert$start) of finishing in a particular end state following a transition from one of three starting states. Numerical values of the mean number of pairwise transitions taken over the four quadrants of the experiment are overlain in (a) and (c), with the corresponding theoretical counts overlain in (b) and (d). Transitions that terminate in a C state have higher probability regardless of start state. (e) Experimental statistical steady-state occupation probabilities for six-particle clusters for $\Gamma/g = (1, 1.05, \ldots, 1.2)$where the points represent the mean $\Pi$ computed over the four quadrants of the experiment. The shaded regions represent the standard error around the mean. Theoretical occupation probabilities predicted from simulations of Eqns. \ref{['eqn:AOUP']} are plotted as a function of the corresponding values of $D$ and $\tau$ determined from the mean-squared displacement and velocity-correlation function (solid dash). (f) Plots of the variation in capillary energy, $U$, when transitioning from a C state through the breaking of a single bond. The lowest activation energies belong to C-C transitions where one or two particles move around the periphery of the cluster in a hinge-like motion, illuminating why the C-C transitions are favored.
• Figure 4: Nonequilibrium thermodynamics and broken detailed balance at the macroscale. (a) The mean experimental entropy production rate $\sigma$ as a function of the vibrational forcing $\Gamma$ (dots) compared to theoretical predictions from Eqns. \ref{['eqn:AOUP']} (solid lines) for $N = 6$ particles. The mean is taken over the four quadrants of the experiment. The parameters $D$ and $\tau$ for the theoretical model are determined from the mean-squared displacement $\sum_{j = 1}^N \langle |\mathbf{x}_j(t) - \mathbf{x}_j(0)|^2 \rangle/N$ and velocity-correlation function $\sum_{j = 1}^N\langle\dot{\mathbf{x}}_j(t)\cdot\dot{\mathbf{x}}_j(t')\rangle/N$, where $\langle\cdot\rangle$ denotes the average over particle trajectories. The shaded region represents the standard error taken over each quadrant of the experiment. (b) Steady-state probability current diagram for the representative case $N = 6$ and $\Gamma = 1.2g$, illustrating broken detailed balance via a persistent flux loop in the direction $\text{T}\rightarrow \text{C}\rightarrow \text{P}$. The quantities $J_{i\rightarrow j} = \Pi_i Q_{ij} - \Pi_j Q_{ji}$ are the net probability currents between microstates $i$ and $j$. (c) Experimental stationary distribution for a 7-particle experiment at $\Gamma = 1.2g$ (green bars) with theoretical predictions of the model \ref{['eqn:AOUP']} shown by the (black) dots. The error bars on top of each probability bar signifies one standard deviation on the mean stationary distribution taken over the four quadrants of the experiment. Inset: steady-state probability current diagram illustrating broken detailed balance via one of the directed cycles $\text{Fl}\rightarrow\text{Tu}\rightarrow\text{Bo}\rightarrow\text{Tr}$.