Oscillatory sheared suspensions, when observed stroboscopically, exhibit a reversible–irreversible transition as a function of the strain amplitude, which is an absorbing phase transition, separating diffusive states on the irreversible side from absorbing states on the reversible side. So far studies of this transition focused on global quantities, e.g., quantifying the irreversibility on the one side of the transition or the time to reach a reversible state on the other side. Here, inspired by depinning-type transitions, we focus on intermittent dynamics close to the transition. We perform simulations of a modified Random Organization Model (ROM), a minimal particle model which we recently adapted to take into account the generic presence of long-range interactions mediated by the fluid, taking the power-law decay exponent α as an additional control parameter of the model. We show that at the absorbing phase transition, this model displays power-law distributed avalanches. We characterize the avalanche statistics in terms of avalanche size, duration and number of particles involved, and we determine the associated exponents. By varying the exponent α , the fractal dimension of avalanches crosses space dimension d, inducing a qualitative change of the spatial structure of avalanches, from compact avalanches when interactions have a short range, to sparse avalanches when interactions are long-ranged. Finally, we characterize the clusters within the avalanches, which we also find to be power-law distributed. Spacetime plots of avalanches of irreversible events (in red) in presence of mediated mechanical noise. Left: long-ranged mechanical noise leads to sparse avalanches. Right: short-ranged mechanical noise leads to compact avalanches.

We investigate homogeneous populations of smart active agents that exchange information with their neighbors to perform a decentralized learning process aimed at achieving a prescribed macroscopic state. Such agents may, for example, represent simple microrobots. The exchanged information comprises tunable parameters governing the agent dynamics, referred to as the individual policy, together with an internal memory encoding previously visited states. This memory is used to evaluate a reward that quantifies the success of a policy to achieve the prescribed state. We extend the kinetic-theory description of collective learning in spatially homogeneous systems [Phys. Rev. Lett. 134, 248302 (2025)] and derive formal evolution equations for the distribution of policies across the population. A central outcome of our theory is the emergence of an effective reward function that fully determines the evolution of the policy distribution and encapsulates the microscopic details of the agents physical and memory dynamics. We obtain closed equations for the policy mean and variance which admit explicit time-dependent solutions under the assumption of Gaussian-distributed memories and polices. To illustrate the framework, we present a series of minimal microscopic models, considering both perfect and partial separation of physical, memory and policy exchange time scales, as well as models with one- and two-dimensional policies. The obtained theoretical results compare well with agent-based numerical simulations. The theory captures key aspects of collective learning, including the influence of population diversity and reward fluctuations on learning performance. Finally, we discuss potential applications to swarm robotics and machine learning, and highlight connections with classical models of biological evolution, including the Replicator equation and the Moran model.

Systems with a complex dynamics like glasses or models of biological evolution are often pictured in terms of a complex landscape, with a large number of possible collective states. We show on the example of a stochastic spin model with nonreciprocal and heterogeneous interactions how the complex landscape picture can be generalized far from equilibrium, where collective states may become time-dependent and exhibit, e.g., spontaneous oscillations, often hidden by the presence of disorder. We identify relevant observables, like the density of entropy production rate, to unveil the spontaneous collective time dependence, and we determine a configurational entropy which counts the number of oscillating collective states when this number grows exponentially with system size.

Systems driven far from equilibrium may exhibit anomalous density fluctuations: active matter with orientational order displays giant density fluctuations at a large scale, while systems of interacting particles close to an absorbing phase transition may exhibit hyperuniformity, suppressing large-scale density fluctuations. We show that these seemingly incompatible phenomena can coexist in nematically ordered active systems, provided activity is conditioned to particle contacts. We characterize this unusual state of matter and unravel the underlying mechanisms simultaneously, leading to spatially enhanced (on large length scales) and suppressed (on intermediate length scales) density fluctuations. Our work highlights the potential for a rich phenomenology in active matter systems in which the particles' activity is triggered by their local environment, and calls for a broader exploration of absorbing phase transitions in orientationally ordered particle systems.

Close to the yielding transition, amorphous solids exhibit a jerky dynamics characterized by plastic avalanches. The statistics of these avalanches have been measured experimentally and numerically using a variety of different triggering protocols, assuming that all of them were equivalent for this purpose. In particular, two main classes of protocols have been studied, deformation under controlled strain or under controlled stress. In this work, we investigate different protocols to generate plasticity avalanches and conduct two-dimensional simulations of an elastoplastic model to examine the protocol dependence of avalanche statistics in yield-stress fluids. We demonstrate that when stress is controlled, the value and even the existence of the exponent governing the probability distribution function of avalanche sizes strongly depend on the protocol chosen to initiate avalanches. This confirms in finite dimension a scenario presented in a previous mean-field analysis. We identify a consistent stress-controlled protocol whose associated avalanches differ from the quasi-static ones in their fractal dimension and dynamical exponent. Remarkably, this protocol also seems to satisfy the scaling relations among exponents previously proposed. Our results underscore the necessity for a cautious interpretation of avalanche universality within elastoplastic models, and more generally within systems where several control parameters exist.

Configurational entropy, or complexity, plays a critical role in characterizing disordered systems such as glasses, yet its measurement often requires significant computational resources. Recently, Rényi entropy, a one-parameter generalization of Shannon entropy, has gained attention across various fields of physics due to its simpler functional form, making it more practical for measurements. In this paper, we compute the Rényi version of complexity for prototypical mean-field disordered models, including the random energy model, its generalization, referred to as the random free energy model, and the p-spin spherical model. We first demonstrate that the Rényi complexity with index m is related to the free energy difference for a generalized annealed Franz-Parisi potential with m clones. Detailed calculations show that for models having one-step replica symmetry breaking (RSB), the Rényi complexity vanishes at the Kauzmann transition temperature T_K, irrespective of m>1, while RSB solutions are required even in the liquid phase. This study strengthens the link between Rényi entropy and the physics of disordered systems and provides theoretical insights for its practical measurements.

Smart active matter has the ability to control its motion guided by individual policies to achieve collective goals. We introduce a theoretical framework to study a decentralized learning process in which agents can locally exchange policies to adapt their behavior and maximize a predefined reward function. We use our formalism to derive explicit hydrodynamic equations for the policy dynamics. We apply the theory to two different microscopic models where policies correspond either to fixed parameters similar to evolutionary dynamics, or to state-dependent controllers known from the field of robotics. We find good agreement between theoretical predictions and agent-based simulations. By deriving fundamental control parameters and uncertainty relations, our work lays the foundations for a statistical physics analysis of decentralized learning.

Driven soft athermal systems may display a reversible-irreversible transition between an absorbing, arrested state and an active phase where a steady-state dynamics sets in. A paradigmatic example consists in cyclically sheared suspensions under stroboscopic observation, for which in absence of contacts during a shear cycle particle trajectories are reversible and the stroboscopic dynamics is frozen, while contacts lead to diffusive stroboscopic motion. The random organization model (ROM), which is a minimal model of the transition, shows a transition which falls into the conserved directed percolation universality class. However, the ROM ignores hydrodynamic interactions between suspended particles, which make contacts a source of long-range mechanical noise that in turn can create new contacts. Here, we generalize the ROM to include long-range interactions decaying like inverse power laws of the distance. Critical properties continuously depend on the decay exponent when it is smaller than the space dimension. Upon increasing the interaction range, the transition turns convex (that is, with an order parameter exponent β > 1 ), fluctuations turn from diverging to vanishing, and hyperuniformity at the transition disappears. We rationalize this critical behavior using a local mean-field model describing how particle contacts are created via mechanical noise, showing that diffusive motion induced by long-range interactions becomes dominant for slowly decaying interactions.

Granular matter like sand is composed of a large number of interacting grains and is, thus, expected to be amenable to a statistical physics treatment. Yet, the frictional properties of grains make the statistical physics of granular matter significantly different from the equilibrium statistical physics of atomic or molecular systems. We use three simple models to illustrate some of the key concepts of the statistical physics introduced by Edwards and co-workers more than 30 years ago to describe shaken granular piles: non-interacting frictional grains attached to a wall by a spring, a chain of frictional grains connected by springs, and a simplified mean-field model of a granular packing. We observe that a chain of frictional grains connected by springs exhibits a critical point at an infinite effective temperature (i.e., infinitely strong shaking) at odds with the zero-temperature critical point generically found in one-dimensional systems at thermal equilibrium in the presence of local interactions.

We study a nonequilibrium ferromagnetic mean-field spin model exhibiting a phase with spontaneous temporal oscillations of the magnetization, on top of the usual paramagnetic and ferromagnetic phases. This behavior is obtained by introducing dynamic field variables coupled to the spins through non-reciprocal couplings. We determine a nonequilibrium generalization of the Landau free energy in terms of the large deviation function of the magnetization and of an appropriately defined smoothed stochastic time derivative of the magnetization. While the transition between paramagnetic and oscillating phase is continuous, the transition between ferromagnetic and oscillating phases is found to be discontinuous, with a coexistence of both phases, one being stable and the other one metastable. Depending on parameter values, the ferromagnetic points may either be inside or outside the limit cycle, leading to different transition scenarios. The stability of these steady states is determined from the large deviation function. We also show that in the coexistence region, the entropy production has a pronounced maximum as a function of system size.

We propose a one-dimensional model of active particles interpolating between quorum sensing models used in the study of motility-induced phase separation (MIPS) and models of congestion of traffic flow on a single-lane highway. Particles have a target velocity with a density-dependent magnitude and a direction that flips with a finite rate that is biased toward moving right. Two key parameters are the bias and the speed relaxation time. MIPS is known to occur in such models at zero bias and zero relaxation time (overdamped dynamics), while a fully biased motion with no velocity reversal models traffic flow on a highway. Using both numerical simulations and continuum equations derived from the microscopic dynamics, we show that a single phase-separated state extends from the usual MIPS to congested traffic flow in the phase diagram defined by the bias and the speed relaxation time. However, in the fully biased case, inertia is essential to observe phase separation, making MIPS and congested traffic flow seemingly different phenomena if not simultaneously considering inertia and tumbling. We characterize the velocity of the dense phase, which is static for usual MIPS and moves backwards in traffic congestion. We also find that in presence of bias, the phase diagram becomes richer, with an additional transition between phase separation and a microphase separation that is seen above a threshold bias or relaxation rate.

The yielding transition in athermal complex fluids can be interpreted as an absorbing phase transition between an elastic, absorbing state with high mesoscopic degeneracy and a flowing, active state. We characterize quantitatively this phase transition in an elastoplastic model under fixed applied shear stress, using a finite-size scaling analysis. We find vanishing critical fluctuations of the order parameter (i.e., the shear rate), and relate this property to the convex character of the phase transition (β>1). We locate yielding within a family of models akin to fixed-energy sandpile (FES) models, only with long-range redistribution kernels with zero modes that result from mechanical equilibrium. For redistribution kernels with sufficiently fast decay, this family of models belongs to a short-range universality class distinct from the conserved directed percolation class of usual FES, which is induced by zero modes.

Experimental measurements of nonlinear dielectric response in glass formers like supercooled glycerol or propylene carbonate have been interpreted as providing evidence for a growing thermodynamic length scale when lowering temperature. A heuristic picture based on coherently flipping "superdipoles" with disordered internal structure has been argued to capture the essence of the experimentally reported behavior, pointing to the key role of effectively disordered interactions in structural glasses. We test these ideas by devising an explicit one-dimensional model of interacting spins incorporating both the spin-glass spirit of the superdipole argument and the necessary long-time decorrelation of structural disorder, encoded here in a slow dynamics of the coupling constants. The frequency-dependent third-order response of the model qualitatively reproduces the typical humped shape reported in experiments. The temperature dependence of the maximum value is also qualitatively reproduced. In contrast, the humped shape of the third-order response is not reproduced by a simple kinetically constrained spin model with noninteracting spins. To rationalize these results, we propose a two-length-scale scenario by distinguishing between the characteristic length of dynamical heterogeneities and a rigidity length that accounts for the local tendency of spins to flip coherently as a block, in the presence of interactions. We show that both length scales are identical in the kinetically constrained spin model, while they have significantly different dynamics in the model of interacting spins.

In a statistical physics context, inverse problems consist of determining microscopic interactions such that a system reaches a predefined collective state. A complex collective state may be prescribed by specifying the overlap distribution between microscopic configurations, a notion originally introduced in the context of disordered systems like spin glasses. We show that in spite of the absence of disorder, nonequilibrium spin models exhibiting spontaneous magnetization oscillations provide a benchmark to prescribe a nontrivial overlap distribution with continuous support, qualitatively analogous to the ones found in disordered systems with full replica symmetry breaking. The overlap distribution can be explicitly tailored to take a broad range of predefined shapes by monitoring the spin dynamics. The presence of a nontrivial overlap distribution is traced back to an average over infinitely many pure states, a feature shared with spin glasses, although the structure of pure states is here much simpler.

We study the onset of collective oscillations at low temperature in a three-dimensional spin model with non-reciprocal short-range interactions. Performing numerical simulations of the model, the presence of a continuous phase transition to global oscillations is confirmed by a finite-size scaling analysis. By systematically varying the interaction range, we show that collective oscillations in this spin model actually result from two successive phase transitions: a mean-field phase transition over finite-size neighborhoods, which leads to the emergence of local noisy oscillators, and a synchronization transition of local noisy oscillators, which generates coherent macroscopic oscillations. Using a Fokker-Planck equation under a local mean-field approximation, we derive from the spin dynamics coupled Langevin equations for the complex amplitudes describing noisy oscillations on a mesoscopic scale. The phase diagram of these coupled equations is qualitatively obtained from a fully-connected (mean-field) approximation. This analytical approach allows us to clearly disentangle the onset of local and global oscillations, and to identify the two main control parameters, expressed as combinations of the microscopic parameters of the spin dynamics, that control the phase diagram of the model.

We introduce a minimal model of energy transfer through scales to describe, at a qualitative level, the subcritical transition between laminar and turbulent flows, viewed in a statistical physics framework as a discontinuous absorbing phase transition. The main control parameter of the model is a Reynolds number that compares energy transfer to viscous dissipation on a large length scale. In spite of its simplicity, the model qualitatively reproduces a number of salient features of the subcritical laminar-turbulent transition, including the existence of an absorbing laminar state, the discontinuous onset of a metastable fluctuating turbulent state above a threshold Reynolds number, and a faster-than-exponential increase of the turbulence lifetime when increasing the Reynolds number. The behavior of the model is also consistent, at high Reynolds number, with the Kolmogorov K41 phenomenology of fully developed turbulence.

In open water, social fish gather to form schools, in which fish generally align with each other. In this work, we study how this social behavior evolves when perturbed by artificial obstacles. We measure the behavior of a group of zebrafish in the presence of a periodic array of pillars. When the pillar density is low, the fish regroup with a typical interdistance and a well-polarized state with parallel orientations, similarly to their behavior in open-water conditions. Above a critical density of pillars, their social interactions, which are mostly based on vision, are screened and the fish spread randomly through the aquarium, orienting themselves along the free axes of the pillar lattice. The abrupt transition from natural to artificial orientation happens when the pillar interdistance is comparable to the social distance of the fish, i.e., their most probable interdistance. We develop a stochastic model of the relative orientation between fish pairs, taking into account alignment, antialignment, and tumbling, from a distribution biased by the environment. This model provides a good description of the experimental probability distribution of the relative orientation between the fish and captures the behavioral transition. Using the model to fit the experimental data provides qualitative information on the evolution of cognitive parameters, such as the alignment or the tumbling rates, as the pillar density increases. At high pillar density, we find that the artificial environment imposes its geometrical constraints to the fish school, drastically increasing the tumbling rate.

We study the effect of introducing separable quenched disorder on a non-equilibrium mean-field spin model exhibiting a phase transition to an oscillating state in the absence of disorder, due to non-reciprocal interactions. In the disordered model, the magnetisation and its time derivative no longer carry the signature of the phase transition to an oscillating state. However, thanks to the separable (Mattis-type) form of the disorder, the presence of oscillations can be revealed by introducing a specific, disorder-dependent observable. We also introduce generalised linear and non-linear susceptibilities associated either with the magnetisation or with its time derivative. While linear susceptibilities show no sign of a phase transition, the third-order susceptibilities present a clear signature of the onset of an oscillating phase. In addition, we show that the overlap distribution also provides evidence for the presence of oscillations, without explicit knowledge of the disorder.

We derive from particle-level dynamics a constitutive model describing the rheology of two-dimensional dense soft suspensions below the jamming transition, in a regime where hydrodynamic interactions between particles are screened. Based on a statistical description of particle dynamics, we obtain through a set of physically plausible approxi-mations a nonlinear tensorial evolution equation for the deviatoric part of the stress tensor, involving the strain rate and vorticity tensors. This tensorial evolution equation involves singular terms usually not taken into account in phenomenological constitutive models, which most often assume a regular expansion in terms of the stress tensor. All coefficients appearing in the equation have known expressions in terms of the microscopic parameters of the model. The predictions of this microscopically grounded constitutive model have several qualitative features that are specific to the rheology of soft suspensions measured in experiments or simulations. The model shows a typical behavior of polymeric viscoelastic materials, such as normal stress differences quadratic in the shear rate gamma, as well as typical behaviors of suspensions of stiff particles, such as a particle pressure linear in gamma and a zero-shear viscosity diverging at the jamming transition. The model also predicts a sharper shear thinning than other viscoelastic models at small shear rates, in qualitative agreement with experimental observations. Furthermore the shear thinning follows a critical scaling close to the jamming transition.

We consider the problem of diffusion with stochastic resetting in a population of random walks where the diffusion coefficient is not constant, but behaves as a power-law of the average resetting rate of the population. Resetting occurs only beyond a threshold distance from the origin. This problem is motivated by physical realizations like soft matter under shear, where diffusion of a walk is induced by resetting events of other walks. We first reformulate in the broader context of diffusion with stochastic resetting the so-called Hébraud–Lequeux model for plasticity in dense soft matter, in which diffusivity is proportional to the average resetting rate. Depending on parameter values, the response to a weak external field may be either linear, or non-linear with a non-zero average position for a vanishing applied field, and the transition between these two regimes may be interpreted as a continuous phase transition. Extending the model by considering a general power-law relation between diffusivity and average resetting rate, we notably find a discontinuous phase transition between a finite diffusivity and a vanishing diffusivity in the small field limit.