Collective motion in large deviations of active particles

Fluctuation Theorems for Heat Exchanges between Passive and Active Baths

In addition to providing general constraints on probability distributions, fluctuation theorems allow us to infer essential information on the role played by temperature in heat exchange phenomena. In this numerical study, we measure the temperature of an out-of-equilibrium active bath using a fluctuation theorem that relates the fluctuations in the heat exchanged between two baths to their temperatures. Our setup consists of a single particle moving between two wells of a quartic potential accommodating two different baths. The heat exchanged between the two baths is monitored according to two definitions: as the kinetic energy carried by the particle whenever it jumps from one well to the other and as the work performed by the particle on one of the two baths when immersed in it. First, we consider two equilibrium baths at two different temperatures and verify that a fluctuation theorem featuring the baths temperatures holds for both heat definitions. Then, we introduce an additional Gaussian coloured noise in one of the baths, so as to make it effectively an active (out-of-equilibrium) bath. We find that a fluctuation theorem is still satisfied with both heat definitions. Interestingly, in this case the temperature obtained through the fluctuation theorem for the active bath corresponds to the kinetic temperature when considering the first heat definition, while it is larger with the second one. We interpret these results by looking at the particle jump phenomenology.

Work Fluctuations for a Harmonically Confined Active Ornstein-Uhlenbeck Particle

We study the active work fluctuations of an active Ornstein-Uhlenbeck particle in the presence of a confining harmonic potential. We tackle the problem analytically both for stationary and generic uncorrelated initial states. Our results show that harmonic confinement can induce singularities in the active work rate function, with linear stretches at large positive and negative active work, at sufficiently large active and harmonic force constants. These singularities originate from big jumps in the displacement and in the active force, occurring at the initial or ending points of trajectories and marking the relevance of boundary terms in this problem.

Spectral signatures of symmetry-breaking dynamical phase transitions

Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. Some of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While macroscopic fluctuation theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete Z_{n} symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. In particular, we show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate subspace. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in several paradigmatic many-body systems, including (1) the one-dimensional boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (2) the three- and four-state Potts model for spin dynamics, which displays discrete rotational symmetry-breaking DPTs for energy fluctuations, and (3) the closed WASEP which presents a continuous symmetry-breaking DPT into a time-crystal phase characterized by a rotating condensate.

Environmental Path-Entropy and Collective Motion

Inspired by the swarming or flocking of animal systems we study groups of agents moving in unbounded 2D space. Individual trajectories derive from a "bottom-up" principle: individuals reorient to maximize their future path entropy over environmental states. This can be seen as a proxy for keeping options open, a principle that may confer evolutionary fitness in an uncertain world. We find an ordered (coaligned) state naturally emerges, as well as disordered states or rotating clusters; similar phenotypes are observed in birds, insects, and fish, respectively. The ordered state exhibits an order-disorder transition under two forms of noise: (i) standard additive orientational noise, applied to the postdecision orientations and (ii) "cognitive" noise, overlaid onto each individual's model of the future paths of other agents. Unusually, the order increases at low noise, before later decreasing through the order-disorder transition as the noise increases further.

Tuning nonequilibrium phase transitions with inertia

In striking contrast to equilibrium systems, inertia can profoundly alter the structure of active systems. Here, we demonstrate that driven systems can exhibit effective equilibrium-like states with increasing particle inertia, despite rigorously violating the fluctuation-dissipation theorem. Increasing inertia progressively eliminates motility-induced phase separation and restores equilibrium crystallization for active Brownian spheres. This effect appears to be general for a wide class of active systems, including those driven by deterministic time-dependent external fields, whose nonequilibrium patterns ultimately disappear with increasing inertia. The path to this effective equilibrium limit can be complex, with finite inertia sometimes acting to accentuate nonequilibrium transitions. The restoration of near equilibrium statistics can be understood through the conversion of active momentum sources to passive-like stresses. Unlike truly equilibrium systems, the effective temperature is now density dependent, the only remnant of the nonequilibrium dynamics. This density-dependent temperature can in principle introduce departures from equilibrium expectations, particularly in response to strong gradients. Our results provide additional insight into the effective temperature ansatz while revealing a mechanism to tune nonequilibrium phase transitions.

Physics-informed graph neural networks enhance scalability of variational nonequilibrium optimal control

When a physical system is driven away from equilibrium, the statistical distribution of its dynamical trajectories informs many of its physical properties. Characterizing the nature of the distribution of dynamical observables, such as a current or entropy production rate, has become a central problem in nonequilibrium statistical mechanics. Asymptotically, for a broad class of observables, the distribution of a given observable satisfies a large deviation principle when the dynamics is Markovian, meaning that fluctuations can be characterized in the long-time limit by computing a scaled cumulant generating function. Calculating this function is not tractable analytically (nor often numerically) for complex, interacting systems, so the development of robust numerical techniques to carry out this computation is needed to probe the properties of nonequilibrium materials. Here, we describe an algorithm that recasts this task as an optimal control problem that can be solved variationally. We solve for optimal control forces using neural network ansätze that are tailored to the physical systems to which the forces are applied. We demonstrate that this approach leads to transferable and accurate solutions in two systems featuring large numbers of interacting particles.

Finite-Time Dynamical Phase Transition in Nonequilibrium Relaxation

We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.

Learning nonequilibrium control forces to characterize dynamical phase transitions

Sampling the collective, dynamical fluctuations that lead to nonequilibrium pattern formation requires probing rare regions of trajectory space. Recent approaches to this problem, based on importance sampling, cloning, and spectral approximations, have yielded significant insight into nonequilibrium systems but tend to scale poorly with the size of the system, especially near dynamical phase transitions. Here we propose a machine learning algorithm that samples rare trajectories and estimates the associated large deviation functions using a many-body control force by leveraging the flexible function representation provided by deep neural networks, importance sampling in trajectory space, and stochastic optimal control theory. We show that this approach scales to hundreds of interacting particles and remains robust at dynamical phase transitions.

Reinforcement learning of rare diffusive dynamics

We present a method to probe rare molecular dynamics trajectories directly using reinforcement learning. We consider trajectories that are conditioned to transition between regions of configuration space in finite time, such as those relevant in the study of reactive events, and trajectories exhibiting rare fluctuations of time-integrated quantities in the long time limit, such as those relevant in the calculation of large deviation functions. In both cases, reinforcement learning techniques are used to optimize an added force that minimizes the Kullback-Leibler divergence between the conditioned trajectory ensemble and a driven one. Under the optimized added force, the system evolves the rare fluctuation as a typical one, affording a variational estimate of its likelihood in the original trajectory ensemble. Low variance gradients employing value functions are proposed to increase the convergence of the optimal force. The method we develop employing these gradients leads to efficient and accurate estimates of both the optimal force and the likelihood of the rare event for a variety of model systems.

Long-ranged correlations in large deviations of local clustering

In systems of diffusing particles, we investigate large deviations of a time-averaged measure of clustering around one particle. We focus on biased ensembles of trajectories, which realize large-deviation events. The bias acts on a single particle, but elicits a response that spans the whole system. We analyze this effect through the lens of macroscopic fluctuation theory, focusing on the coupling of the bias to hydrodynamic modes. This explains that the dynamical free energy has nontrivial scaling relationships with the system size, in 1 and 2 spatial dimensions. We show that the long-ranged response to a bias on one particle also has consequences when biasing two particles.

Phase Diagram of Active Brownian Spheres: Crystallization and the Metastability of Motility-Induced Phase Separation

Motility-induced phase separation (MIPS), the phenomenon in which purely repulsive active particles undergo a liquid-gas phase separation, is among the simplest and most widely studied examples of a nonequilibrium phase transition. Here, we show that states of MIPS coexistence are in fact only metastable for three-dimensional active Brownian particles over a very broad range of conditions, decaying at long times through an ordering transition we call active crystallization. At an activity just above the MIPS critical point, the liquid-gas binodal is superseded by the crystal-fluid coexistence curve, with solid, liquid, and gas all coexisting at the triple point where the two curves intersect. Nucleating an active crystal from a disordered fluid, however, requires a rare fluctuation that exhibits the nearly close-packed density of the solid phase. The corresponding barrier to crystallization is surmountable on a feasible timescale only at high activity, and only at fluid densities near maximal packing. The glassiness expected for such dense liquids at equilibrium is strongly mitigated by active forces, so that the lifetime of liquid-gas coexistence declines steadily with increasing activity, manifesting in simulations as a facile spontaneous crystallization at extremely high activity.