Extreme synchronization transitions

Standard epidemic models exhibit one continuous, second order phase transition to macroscopic outbreaks. However, interventions to control outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such interventions modify the type of phase transition. In particular, we uncover three distinct types of explosive phase transitions for epidemic dynamics with capacity-limited interventions. Depending on the capacity limit, interventions may (i) leave the standard second order phase transition unchanged but exponentially suppress the probability of large outbreaks, (ii) induce a first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a secondary explosive yet continuous third-order transition. These insights highlight inherent limitations in predicting and containing epidemic outbreaks. More generally our study offers a cornerstone example of a third-order explosive phase transition in complex systems.

In recent studies, it has been established that higher-order interactions in coupled oscillators can induce a process from continuous to explosive phase transition. In this study, we identify a phase transition, termed the stepwise explosive phase transition, characterized by the emergence of multiple critical phase plateaus in a globally frequency-weighted coupled pendulum model. This transition bridges the continuous and explosive phase transitions, arising from a delicate balance between attractive higher-order interactions and repulsive pairwise interactions. Specifically, the stepwise explosive phase transition occurs when the higher-order coupling is moderate, neither large nor small, while the pairwise interactions remain repulsive. Our analysis shows that stronger attractive higher-order interactions necessitate weaker repulsive pairwise interactions, leading to partial frequency locking among oscillators and triggering the stepwise transition. We construct an analytical framework using self-consistent equationsto provide an approximation of the steady-state behavior. This study uncovers an alternative pathway to the desynchronization, and it provides additional insights into the phase transitions in coupled dynamical networks.

Based on quasistationary distribution ideas, a general finite size scaling theory is proposed for discontinuous nonequilibrium phase transitions into absorbing states. Analogously to the equilibrium case, we show that quantities such as response functions, cumulants, and equal area probability distributions all scale with the volume, thus allowing proper estimates for the thermodynamic limit. To illustrate these results, five very distinct lattice models displaying nonequilibrium transitions-to single and infinitely many absorbing states-are investigated. The innate difficulties in analyzing absorbing phase transitions are circumvented through quasistationary simulation methods. Our findings (allied to numerical studies in the literature) strongly point to a unifying discontinuous phase transition scaling behavior for equilibrium and this important class of nonequilibrium systems.

Recently, the learning by confusion (LbC) approach has been proposed as a machine learning tool to determine the critical temperature T_{c} of phase transitions without any prior knowledge of its even approximate value. The method has been proven effective, but it has been used only for continuous phase transitions, where the confusion results only from deliberate incorrect labeling of the data. However, in the case of a discontinuous phase transition, additional confusion can result from the coexistence of different phases. To verify whether the confusion scheme can also be used for discontinuous phase transitions, we apply the LbC method to three microscopic models, the Blume-Capel, the q-state Potts, and the Falicov-Kimball models, which undergo continuous or discontinuous phase transitions depending on model parameters. With the help of a simple model, we predict that the phase coexistence present in discontinuous phase transitions can indeed make the neural network more confused and thus decrease its performance. However, numerical calculations performed for the models mentioned above indicate that other aspects of this kind of phase transition are more important and can render the LbC method even less effective. Nevertheless, we demonstrate that in some cases the same aspects allow us to use the LbC method to identify the order of a phase transition.

We have carried out analytical and numerical studies of the spin-boson model in the sub-ohmic regime with the influence of both the diagonal and the off-diagonal coupling accounted for, via the Davydov D1 variational ansatz. While a second-order phase transition is known to be exhibited by this model in the presence of diagonal coupling only, we demonstrate the emergence of a discontinuous first order phase transition upon incorporation of the off-diagonal coupling. A plot of the ground state energy versus magnetization highlights the discontinuous nature of the transition between the isotropic (zero magnetization) state and nematic (finite magnetization) phases. We have also calculated the entanglement entropy and a discontinuity found at a critical coupling strength further supports the discontinuous crossover in the spin-boson model in the presence of off-diagonal coupling. It is further revealed via a canonical transformation approach that for the special case of identical exponents for the spectral densities of the diagonal and the off-diagonal coupling, there exists a continuous crossover from a single localized phase to doubly degenerate localized phase with differing magnetizations.

We define the characteristic temperature T C of a material by using the behaviour of the specific heat in the vicinity of this temperature: C p,N (T ) = f(1 − (T/T C ),p,N ) for T < T C and C p,N (T ) = g((T/T C ) − 1,p,N ) for T > T C where p is the external pressure and N is the number of particles. The characteristic temperature T C = T C ( p, N ) can e.g. be identified with the transition temperature in the case of continuous phase transitions or with one of the two characteristic temperatures (on heating) and (on cooling) in the case of discontinuous phase transitions. Using the idea of the characteristic temperature, we present a new method on how to calculate the Gibbs potential G = G(T, p, N) in the vicinity of the characteristic temperature, without it being a necessity to know the explicite expressions for f and g. This general and uniform approach allows to derive many thermodynamical relations connecting measurable quantities at the characterisic point as e.g. Clausius–Clapeyron equation and its analogs, valid for discontinuous phase transitions and 15 relations valid for both continuous and discontinuous phase transitions, 6 of them are new. As a special case of these latter relations, we find six relations valid for continuous phase transitions (Ehrenfest equation and its analogs). The derived relations result exclusively from the definition of the characteristic point and are independent from the underlying microscopical mechanism leading to phase transitions (as e.g. solid–liquid, liquid–vapour, magnetic, ferroelectric, structural and superconducting transitions) and therefore, they are generally valid. These relations can be very important for material science. Using some of them it is possible to predict (without performing additional experiments) whether the transition or characteristic temperature T C of a given material sample will increase or decrease under doping (injection) or applied pressure. Similar 24 thermodynamical relations can also be found with ease for systems described by the Helmholtz potential F = F(T,V, N ).

A technique based on steady-state fluorescence measurements is introduced to study the discontinuous volume phase transition of hydrogen-bonding gel. The fluorescence light intensity of pyranine (1-Hydroxypyrene-3, 6, 8-trisulfonic acid) bonded to the gel by means of methacryl amino-propyl-trimethyl ammonium chloride was monitored during the swelling of poly(methacrylic acid-co-dimethyl acrylamide) copolymer in water as a function of temperature. A discontinuous volume phase transition around 60 degrees C and an additional phase around 30 degrees C, which may be interpreted as the freely fluctuating phase in a collapsed state, were observed when the fluorescence intensity changes.

The ɛ-susceptible-infected-susceptible (SIS) epidemic model on a graph adds an independent, Poisson self-infection process with rate ɛ to the "classical" Markovian SIS process. The steady state in the classical SIS process (with ɛ=0) on any finite graph is the absorbing or overall-healthy state, in which the virus is eradicated from the network. We report that there always exists a phase transition around τ_{c}^{ɛ}=O(ɛ^{-1/N-1}) in the ɛ-SIS process on the complete graph K_{N} with N nodes, above which the effective infection rate τ>τ_{c}^{ɛ} causes the average steady-state fraction of infected nodes to approach that of the mean-field approximation, no matter how small, but not zero, the self-infection rate ɛ is. For τ<τ_{c}^{ɛ} and small ɛ, the network is almost overall healthy. The observation was found by mathematical analysis on the complete graph K_{N}, but we claim that the phase transition of explosive type may also occur in any other finite graph. We thus conclude that the overall-healthy state of the classical Markovian SIS model is unstable in the ɛ-SIS process and, hence, unlikely to exist in reality, where "background" infection ɛ>0 is imminent.

Topological states of matter are characterized by global topological invariants which change their value across a topological quantum phase transition. It is commonly assumed that the transition between topologically distinct noninteracting gapped phases of fermions is necessarily accompanied by the closing of the band gap as long as the symmetries of the system are maintained. We show that such a quantum phase transition is possible without closing the gap in the case of a three-dimensional topological band insulator. We demonstrate this by calculating the free energy of the minimal model for a topological insulator, the Bernevig-Hughes-Zhang model, and show that as the band curvature continuously varies, a jump between the band gap minima corresponding to the topologically trivial and nontrivial insulators occurs. Therefore, this first order phase transition is a generic feature of three-dimensional topological band insulators. For a certain parameter range we predict a re-entrant topological phase transition. We discuss our findings in connection with the recent experimental observation of a discontinuous topological phase transition in a family of topological crystalline insulators.

High-order network models have emerged as powerful frameworks for modeling complex dynamical systems that involve multi-body interactions in recent years. Despite their widespread adoption in prior studies, the subtle difference between high-order structures, even those of the same order, has frequently been overlooked. To fill this gap in between, this study takes four typical triadic (second-order) configurations as examples and examines their influence on complex contagion process described by the Susceptible-Infected-Susceptible (SIS) model. A network construction model is first proposed to generate high-order networks with one of the four triadic configurations while maintaining the average second-order degree as well as the first-order structure. Numerical simulations of contagions on these networks reveal that, while different triadic configurations do not alter the qualitative characteristics of the phase transition (i.e., from continuous to discontinuous phase transitions) in contagion processes, they have a noticeable impact on the second-order epidemic threshold for discontinuous phase transitions. Specifically, networks with 2-simplices exhibit the smallest threshold for discontinuous phase transitions, whereas those with 3-cycle structures display the largest threshold. By employing the heterogeneous mean-field (HMF) approach, the critical second-order epidemic threshold for the emergence of discontinuous phase transitions is analytically derived, providing a theoretical explanation for the numerical findings.&amp;#xD;

Lateral predictive coding is a recurrent neural network that creates energy-efficient internal representations by exploiting statistical regularity in sensory inputs. Here, we analytically investigate the trade-off between information robustness and energy in a linear model of lateral predictive coding and numerically minimize a free energy quantity. We observed several phase transitions in the synaptic weight matrix, particularly a continuous transition that breaks reciprocity and permutation symmetry and builds cyclic dominance and a discontinuous transition with the associated sudden emergence of tight balance between excitatory and inhibitory interactions. The optimal network follows an ideal gas law over an extended temperature range and saturates the efficiency upper bound of energy use. These results provide theoretical insights into the emergence and evolution of complex internal models in predictive processing systems.

An infinite array of globally coupled overdamped constituents moving in a double-well potential with nth order saturation term under the influence of additive Gaussian white noise is investigated. The system exhibits a continuous phase transition from a symmetric phase to a symmetry-broken phase. The qualitative behavior is independent on n. The critical point is calculated for strong and for weak noise; these limits are also bounds for the critical point. Introducing an additional nonlinearity, such that the potential can have up to three minima, leads to richer behavior. There the parameter space divides into three regions: a region with a symmetric phase, a region with a phase of broken symmetry and a region where both phases coexist. The region of coexistence collapses into one of the others via a discontinuous phase transition, whereas the transition between the symmetric phase and the phase of broken symmetry is continuous. The tricritical point where the three regions intersect can be calculated for strong and for weak noise. These limiting values form tight bounds on the tricritical point. In the region of coexistence simulations of finite systems are performed. One finds that the stationary distribution of finite but large systems differs qualitatively from the one of the infinite system. Hence the limits of stationarity and large system size do not commute.

Neutron Diffraction Study of the Low-Temperature Phases of Sodium Oxide Nitrite, Na3ONO2

We present a comprehensive structural and thermomechanical investigation of N-salicylideneaniline, a Schiff base derivative that exhibits remarkable thermosalient phase transition behavior. By combining variable-temperature X-ray powder diffraction (VT-XRPD), differential scanning calorimetry (DSC), hot-stage microscopy, and Hirshfeld surface analysis, we reveal two distinct thermosalient mechanisms operating in different polymorphic forms. Form I displays pronounced anisotropic thermal expansion with negative strain along a principal axis, culminating in a sudden and explosive phase transition into Form IV. In contrast, Form III transforms more gradually through a microstrain accumulation mechanism. Fingerprint plots and contact evolution from Hirshfeld surface analysis further support this dual-mechanism model. These insights highlight the importance of integrating macro- and microscale structural descriptors to fully capture the mechanical behavior of responsive molecular solids. The findings not only enhance the fundamental understanding of thermosalience but also inform the rational design of functional materials for actuating and sensing applications.

There are conflicting reports in the literature regarding the finite-size scaling of the Liouvillian gap and dynamical fluctuations at discontinuous phase transitions, with various studies reporting either exponential or power-law behavior. We clarify this issue by employing large deviation theory. We distinguish two distinct classes of discontinuous phase transitions that have different dynamical properties. The first class is associated with phase coexistence, i.e., the presence of multiple stable attractors of the system dynamics (e.g., local minima of the free-energy functional) in a finite phase diagram region around the phase transition point. In that case, one observes asymptotic exponential scaling related to stochastic switching between attractors (though the onset of exponential scaling may sometimes occur for very large system sizes). In the second class, there is no phase coexistence away from the phase transition point, while at the phase transition point itself there are infinitely many attractors. In that case, one observes power-law scaling related to the diffusive nature of the system relaxation to the stationary state.