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We provide a stochastic thermodynamic description across scales for N identical units with all-to-all interactions that are driven away from equilibrium by different reservoirs and external forces. We start at the microscopic level with Poisson rates describing transitions between many-body states. We then identify an exact coarse graining leading to a mesoscopic description in terms of Poisson transitions between system occupations. We proceed studying macroscopic fluctuations using the Martin-Siggia-Rose formalism and large deviation theory. In the macroscopic limit (N → ∞), we derive the exact nonlinear (mean-field) rate equation describing the deterministic dynamics of the most likely occupations. We identify the scaling of the energetics and kinetics ensuring thermodynamic consistency (including the detailed fluctuation theorem) across microscopic, mesoscopic and macroscopic scales. The conceptually different nature of the 'Shannon entropy' (and of the ensuing stochastic thermodynamics) at different scales is also outlined. Macroscopic fluctuations are calculated semi-analytically in an out-of-equilibrium Ising model. Our work provides a powerful framework to study thermodynamics of nonequilibrium phase transitions.

Tipping points in complex systems may imply risks of unwanted collapse, but also opportunities for positive change. Our capacity to navigate such risks and opportunities can be boosted by combining emerging insights from two unconnected fields of research. One line of work is revealing fundamental architectural features that may cause ecological networks, financial markets, and other complex systems to have tipping points. Another field of research is uncovering generic empirical indicators of the proximity to such critical thresholds. Although sudden shifts in complex systems will inevitably continue to surprise us, work at the crossroads of these emerging fields offers new approaches for anticipating critical transitions.

Understanding nonequilibrium systems and the consequences of irreversibility for the system's behavior as compared to the equilibrium case, is a fundamental question in statistical physics. Here, we investigate two types of nonequilibrium phase transitions, a second-order and an infinite-order phase transition, in a prototypical q-state vector Potts model which is driven out of equilibrium by coupling the spins to heat baths at two different temperatures. We discuss the behavior of the quantities that are typically considered in the vicinity of (equilibrium) phase transitions, like the specific heat, and moreover investigate the behavior of the entropy production (EP), which directly quantifies the irreversibility of the process. For the second-order phase transition, we show that the universality class remains the same as in equilibrium. Further, the derivative of the EP rate with respect to the temperature diverges with a power-law at the critical point, but displays a non-universal critical exponent, which depends on the temperature difference, i.e., the strength of the driving. For the infinite-order transition, the derivative of the EP exhibits a maximum in the disordered phase, similar to the specific heat. However, in contrast to the specific heat, whose maximum is independent of the strength of the driving, the maximum of the derivative of the EP grows with increasing temperature difference. We also consider entropy fluctuations and find that their skewness increases with the driving strength, in both cases, in the vicinity of the second-order transition, as well as around the infinite-order transition.

Stories of mega-jams that last tens of hours or even days appear not only in fiction but also in reality. In this context, it is important to characterize the collapse of the network, defined as the transition from a characteristic travel time to orders of magnitude longer for the same distance traveled. In this multicity study, we unravel this complex phenomenon under various conditions of demand and translate it to the travel time of the individual drivers. First, we start with the current conditions, showing that there is a characteristic time τ that takes a representative group of commuters to arrive at their destinations once their maximum density has been reached. While this time differs from city to city, it can be explained by Γ, defined as the ratio of the vehicle miles traveled to the total vehicle distance the road network can support per hour. Modifying 􀀀 can improve τ and directly inform planning and infrastructure interventions. In this study we focus on measuring the vulnerability of the system by increasing the volume of cars in the network, keeping the road capacity and the empirical spatial dynamics from origins to destinations unchanged. We identify three states of urban traffic, separated by two distinctive transitions. The first one describes the appearance of the first bottlenecks and the second one the collapse of the system. This collapse is marked by a given number of commuters in each city and it is formally characterized by a nonequilibrium phase transition.

Significance Catastrophic shifts such as desertification processes, massive extinctions, or stock market collapses are ubiquitous threats in nature and society. In these events, there is a shift from one steady state to a radically different one, from which recovery is exceedingly difficult. Thus, there is a huge interest in predicting and eventually preventing catastrophic shifts. Here we explore the influence of key mechanisms such as demographic fluctuations, heterogeneity, and diffusion, which appear generically in real circumstances. The mechanisms we study could ideally be exploited to smooth abrupt shifts and to make transitions progressive and easier to revert. Thus, our findings could be of potential importance for ecosystem management and biodiversity conservation. Transitions between regimes with radically different properties are ubiquitous in nature. Such transitions can occur either smoothly or in an abrupt and catastrophic fashion. Important examples of the latter can be found in ecology, climate sciences, and economics, to name a few, where regime shifts have catastrophic consequences that are mostly irreversible (e.g., desertification, coral reef collapses, and market crashes). Predicting and preventing these abrupt transitions remains a challenging and important task. Usually, simple deterministic equations are used to model and rationalize these complex situations. However, stochastic effects might have a profound effect. Here we use 1D and 2D spatially explicit models to show that intrinsic (demographic) stochasticity can alter deterministic predictions dramatically, especially in the presence of other realistic features such as limited mobility or spatial heterogeneity. In particular, these ingredients can alter the possibility of catastrophic shifts by giving rise to much smoother and easily reversible continuous ones. The ideas presented here can help further understand catastrophic shifts and contribute to the discussion about the possibility of preventing such shifts to minimize their disruptive ecological, economic, and societal consequences.

We examine skyrmions driven periodically over random quenched disorder and show that there is a transition from reversible motion to a state in which the skyrmion trajectories are chaotic or irreversible. We find that the characteristic time required for the system to organize into a steady reversible or irreversible state exhibits a power law divergence near a critical ac drive period, with the same exponent as that observed for reversible to irreversible transitions in periodically sheared colloidal systems, suggesting that the transition can be described as an absorbing phase transition in the directed percolation universality class. We compare our results to the behavior of an overdamped system and show that the Magnus term enhances the irreversible behavior by increasing the number of dynamically accessible orbits. We discuss the implications of this work for skyrmion applications involving the long time repeatable dynamics of dense skyrmion arrays.

We identify a new class of phase transitions when calculating the Hall conductance of two-band Chern insulators in the long-time limit after a global quench of the Hamiltonian. The Hall conductance is expressed as the integral of the Berry curvature in the diagonal ensemble. Even if the Chern number of the unitarily-evolving wave function is conserved, the Hall conductance as a function of the energy gap in the post-quench Hamiltonian displays a continuous but nonanalytic behavior, that is it has a logarithmically divergent derivative as the gap closes. The coefficient of this logarithmic function is the ratio of the change of the Chern number for the ground state of the post-quench Hamiltonian to the energy gap in the initial state. This nonanalytic behavior is universal in two-band Chern insulators.

We study the Galam’s majority-rule model in the presence of an independent behavior that can be driven intrinsically or can be mediated by information regarding the collective opinion of the whole population. We first apply the mean-field approach where we obtained an explicit time-dependent solution for the order parameter of the model. We complement our results with Monte Carlo simulations where our findings indicate that independent opinion leads to order–disorder continuous nonequilibrium phase transitions. Finite-size scaling analysis show that the model belongs to the mean-field Ising model universality class. Moreover, results from an approach with the Kramers–Moyal coefficients provide insights about the social volatility.

We show that both square and kagome artificial spin ice systems exhibit disorder-induced nonequilibrium phase transitions, with power law avalanche distributions at the critical disorder level. The different nature of geometrical frustration in the two lattices produces distinct types of critical avalanche behavior. For the square ice, the avalanches involve the propagation of locally stable domain walls separating the two polarized ground states, and the scaling collapse agrees with an interface depinning mechanism. In contrast, avalanches in the fully frustrated kagome ice exhibit pronounced branching behaviors that resemble those found in directed percolation. The kagome ice also shows an interesting crossover in the power-law scaling of the avalanches at low disorder. Our results show that artificial spin ices are ideal systems in which to study nonequilibrium critical point phenomena.

We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka–Volterra type interactions defined on a d-dimensional lattice. Introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical, deterministic mean-field picture. Already within mean-field theory, however, spatial constraints, modeling locally limited resources, lead to the emergence of a continuous active-to-absorbing state phase transition. Field-theoretic arguments, supported by Monte Carlo simulation results, indicate that this transition, which represents an extinction threshold for the predator population, is governed by the directed percolation universality class. In the active state, where predators and prey coexist, the classical center singularities with associated population cycles are replaced by either nodes or foci. In the vicinity of the stable nodes, the system is characterized by essentially stationary localized clusters of predators in a sea of prey. Near the stable foci, however, the stochastic lattice Lotka–Volterra system displays complex, correlated spatio-temporal patterns of competing activity fronts. Correspondingly, the population densities in our numerical simulations turn out to oscillate irregularly in time, with amplitudes that tend to zero in the thermodynamic limit. Yet in finite systems these oscillatory fluctuations are quite persistent, and their features are determined by the intrinsic interaction rates rather than the initial conditions. We emphasize the robustness of this scenario with respect to various model perturbations.