Directed Percolation Universality Class Research Articles

Phase Transition to Turbulence via Moving Fronts.

Directed percolation (DP), a universality class of continuous phase transitions, has recently been established as a possible route to turbulence in subcritical wall-bounded flows. In canonical straight pipe or planar flows, the transition occurs via discrete large-scale turbulent structures, known as puffs in pipe flow or bands in planar flows, which either self-replicate or laminarize. However, these processes might not be universal to all subcritical shear flows. Here, we design a numerical experiment that eliminates discrete structures in plane Couette flow and show that it follows a different, simpler transition scenario: turbulence proliferates via expanding fronts and decays via spontaneous creation of laminar zones. We map this phase transition onto a stochastic one-variable system. The level of turbulent fluctuations dictates whether moving-front transition is discontinuous, or continuous and within the DP universality class, with profound implications for other hydrodynamic systems.

Topology-dependent coalescence controls scaling exponents in finite networks

Studies of neural avalanches across different data modalities led to the prominent hypothesis that the brain operates near a critical point. The observed exponents often indicate the mean-field directed-percolation universality class, leading to the fully connected or random network models to study the avalanche dynamics. However, cortical networks have distinct nonrandom features and spatial organization that is known to affect critical exponents. Here we show that distinct empirical exponents arise in networks with different topology and depend on the network size. In particular, we find apparent scale-free behavior with mean-field exponents appearing as quasicritical dynamics in structured networks. This quasicritical dynamics cannot be easily discriminated from an actual critical point in small networks. We find that the local coalescence in activity dynamics can explain the distinct exponents. Therefore, both topology and system size should be considered when assessing criticality from empirical observables. Published by the American Physical Society 2024

Absorbing state phase transition with Clifford circuits

The role of quantum fluctuations in modifying the critical behavior of nonequilibrium phase transitions is a fundamental but unsolved question. In this study, we examine the absorbing state phase transition of a 1D chain of qubits undergoing a contact process that involves both coherent and classical dynamics. We adopt a discrete-time quantum model with states that can be described in the stabilizer formalism, and therefore allows for an efficient simulation of large system sizes. The extracted critical exponents indicate that the absorbing state phase transition of this Clifford circuit model belongs to the directed percolation universality class. This suggests that the inclusion of quantum fluctuations does not necessarily alter the critical behavior of nonequilibrium phase transitions of purely classical systems. Finally, we extend our analysis to a non-Clifford circuit model, where a tentative scaling analysis in small systems reveals critical exponents that are also consistent with the directed percolation universality class. Published by the American Physical Society 2024

Synchronization transitions in coupled q-deformed logistic maps

The concept of q-deformation has been expanded in a variety of situations including q-deformed nonlinear maps. Coupled q-deformed logistic maps with positive and negative q-diffusive coupling are investigated. The transition to state of synchronized fixed point in the thermodynamic limit for random initial conditions is studied as dynamic phase transitions. Though rare for one-dimensional maps, q-deformed maps show multistability. For weak coupling, the synchronized state may not be observed in the thermodynamic limit with random initial conditions even if it is linearly stable. Thus multistability persists. However, for large lattices with random initial conditions, five well-defined critical points are observed where the transition to a synchronized state occurs. Two of these show continuous phase transition. One of them is in the directed percolation universality class. In another case, the order parameter shows power-law decay superposed with oscillations on a logarithmic scale at the critical point and does not match with known universality classes.

Transition to period-3 synchronized state in coupled gauss maps.

We study coupled Gauss maps in one dimension with nearest-neighbor interactions. We observe transitions from spatiotemporal chaos to period-3 states in a coarse-grained sense and synchronized period-3 states. Synchronized fixed points are frequently observed in one dimension. However, synchronized periodic states are rare. The obvious reason is that it is very easy to create defects in one dimension. We characterize all transitions using the following order parameter. Let x∗ be the fixed point of the map. The values above (below) x∗ are classified as +1 (-1) spins. We expect all sites to return to the same band after three time steps for a coarse-grained periodic or three-period state. We define the flip rate F(t) as the fraction of sites i such that si(3t-3)≠si(t). It is zero in the coarse-grained periodic state. This state may or may not be synchronized. We observe three different transitions. (a) If different sites reach different bands, the transition is in the directed-percolation universality class. (b) If all sites reach the same band, we find an Ising-type transition. (c) A synchronized period-3 state where a new exponent is observed. We also study the finite-size scaling at critical points. The exponents obtained indicate that the synchronized period-3 transition is in a new universality class.

Criticality of neuronal avalanches in human sleep and their relationship with sleep macro- and micro-architecture

Sleep plays a key role in preserving brain function, keeping brain networks in a state that ensures optimal computation. Empirical evidence indicates that this state is consistent with criticality, where scale-free neuronal avalanches emerge. However, the connection between sleep architecture and brain tuning to criticality remains poorly understood. Here, we characterize the critical behavior of avalanches and study their relationship with sleep macro- and micro-architectures, in particular, the cyclic alternating pattern (CAP). We show that avalanches exhibit robust scaling behaviors, with exponents obeying scaling relations consistent with the mean-field directed percolation universality class. We demonstrate that avalanche dynamics is modulated by the NREM-REM cycles and that, within NREM sleep, avalanche occurrence correlates with CAP activation phases-indicating a potential link between CAP and brain tuning to criticality. The results open new perspectives on the collective dynamics underlying CAP function, and on the relationship between sleep architecture, avalanches, and self-organization to criticality.

Random organization and non-equilibrium hyperuniform fluids on a sphere.

Randomly organizing hyperuniform fluid induced by reciprocal activation is a non-equilibrium fluid with vanishing density fluctuations at large length scales such as crystals. Here, we extend this new state of matter to a closed manifold, namely a spherical surface. We find that the random organization on a spherical surface behaves similar to that in two dimensional Euclidean space, and the absorbing transition on a sphere also belongs to the conserved directed percolation universality class. Moreover, the reciprocal activation can also induce a non-equilibrium hyperuniform fluid on a sphere. The spherical structure factor at the absorbing transition and the non-equilibrium hyperuniform fluid phases are scaled as S(l → 0) ∼ (l/R)0.45 and S(l → 0) ∼ l(l + 1)/R2, respectively, which are both hyperuniform according to the definition of hyperuniformity on a sphere with l, the wave number, and R, the radius of the spherical surface. We also consider the impact of inertia in realistic hyperuniform fluids, and it is found only by adding an extra length-scale, above which hyperuniform scaling appears. Our finding suggests a new method for creating non-equilibrium hyperuniform fluids on closed manifolds to avoid boundary effects.

Tricritical behavior in epidemic dynamics with vaccination

We scrutinize the phenomenology arising from a minimal vaccination-epidemic (MVE) dynamics using three methods: mean-field approach, Monte Carlo simulations, and finite-size scaling analysis. The mean-field formulation reveals that the MVE model exhibits either a continuous or a discontinuous active-to-absorbing phase transition, accompanied by bistability and a tricritical point. However, on square lattices, we detect no signs of bistability, and we disclose that the active-to-absorbing state transition has a scaling invariance and critical exponents compatible with the continuous transition of the directed percolation universality class. Additionally, our findings indicate that the tricritical and crossover behaviors of the MVE dynamics belong to the universality class of mean-field tricritical directed percolation.

Universal crossover in quantum circuits governed by a proximate classical error correction transition

We formulate a semi-classical circuit model to clarify the role of quantum entanglement in the recently discovered encoding phase transitions in quantum circuits with measurements. As a starting point we define a random circuit model with nearest neighbor classical gates interrupted by erasure errors. In analogy with the quantum setting, this system undergoes a purification transition at a critical error rate above which the classical information entropy in the output state vanishes. We show that this phase transition is in the directed percolation universality class, consistent with the fact that having zero entropy is an absorbing state of the dynamics; this classical circuit cannot generate entropy. Adding an arbitrarily small density of quantum gates in the presence of errors eliminates the transition by destroying the absorbing state: the quantum gates generate internal entanglement, which can be effectively converted to classical entropy by the errors. We describe the universal properties of this instability in an effective model of the semi-classical circuit. Our model highlights the crucial differences between information dynamics in classical and quantum circuits.

Constrained Dynamics and Directed Percolation

In a recent work [A. Deger et al., Phys. Rev. Lett. 129, 160601 (2022).PRLTAO0031-900710.1103/PhysRevLett.129.160601] we have shown that kinetic constraints can completely arrest many-body chaos in the dynamics of a classical, deterministic, translationally invariant spin system with the strength of the constraint driving a dynamical phase transition. Using extensive numerical simulations and scaling analyses we demonstrate here that this constraint-induced phase transition lies in the directed percolation universality class in both one and two spatial dimensions.

Size-dependent transient nature of localized turbulence in transitional channel flow

It has been reported that a fully localized turbulent band in channel flow becomes sustained when the Reynolds number is above a threshold. Here we show evidence that turbulent bands are of a transient nature instead. When the band length is controlled to be fixed, the lifetime of turbulent bands appears to be stochastic and exponentially distributed, a sign of a memoryless transient nature. Besides increasing with the Reynolds number, the mean lifetime also strongly increases with the band length. Given that the band length always changes over time in real channel flow, this size dependence may translate into a time dependence, which needs to be taken into account when clarifying the relationship between channel flow transition and the directed percolation universality class.

Contact process with simultaneous spatial and temporal disorder

We study the absorbing-state phase transition in the one-dimensional contact process under the combined influence of spatial and temporal random disorders. We focus on situations in which the spatial and temporal disorders decouple. Couched in the language of epidemic spreading, this means that some spatial regions are, at all times, more favorable than others for infections, and some time periods are more favorable than others independent of spatial location. We employ a generalized Harris criterion to discuss the stability of the directed percolation universality class against such disorder. We then perform large-scale Monte Carlo simulations to analyze the critical behavior in detail. We also discuss how the Griffiths singularities that accompany the nonequilibrium phase transition are affected by the simultaneous presence of both disorders.

Mechanism for turbulence proliferation in subcritical flows

The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here, we propose the first-ever dynamical mechanism for puff proliferation—the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two-puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall-bounded flows, and implications for the universality of the directed percolation picture, are discussed.

Measurement-induced phase transition in a chaotic classical many-body system

Local measurements in quantum systems are projective operations which act to counteract the spread of quantum entanglement. Recent work has shown that local, random measurements applied to a generic volume-law entanglement generating many-body system are able to force a transition into an area-law phase. This work shows that projective operations can also force a similar classical phase transition; we show that local projections in a chaotic system can freeze information dynamics. In rough analogy with measurement-induced phase transitions, this is characterized by an absence of information spreading instead of entanglement entropy. We leverage a damage-spreading model of the classical transition to predict the butterfly velocity of the system both near to and away from the transition point. We map out the full phase diagram and show that the critical point is shifted by local projections, but remains in the directed percolation universality class. We discuss the implication for other classical chaotic many-body systems.

Kibble-Zurek mechanism for nonequilibrium phase transitions in driven systems with quenched disorder

We show that the Kibble-Zurek mechanism applies to nonequilibrium phase transitions found in driven assemblies of superconducting vortices and colloids moving over quenched disorder where a transition occurs from a plastic disordered flowing state to a moving anisotropic crystal. We measure the density of topological defects as a function of quench rate through the nonequilibrium phase transition, and find that on the ordered side of the transition, the topological defect density ρd scales as a power law, {rho }_{{{{{{{{rm{d}}}}}}}}}propto 1/{t}_{{{{{{{{rm{q}}}}}}}}}^{beta }, where tq is the quench time duration, consistent with the Kibble-Zurek mechanism. We show that scaling with the same exponent holds for varied strengths of quenched disorder and that the exponents fall in the directed percolation universality class. Our results suggest that the Kibble-Zurek mechanism can be applied to the broader class of systems that exhibit absorbing phase transitions.

Directed percolation in temporal networks

Connectivity and reachability on temporal networks, which can describe the spreading of a disease, decimation of information or the accessibility of a public transport system over time, have been among the main contemporary areas of study in complex systems for the last decade. However, while isotropic percolation theory successfully describes connectivity in static networks, a similar description has not been yet developed for temporal networks. Here address this problem and formalize a mapping of the concept of temporal network reachability to percolation theory. We show that the limited-waiting-time reachability, a generic notion of constrained connectivity in temporal networks, displays directed percolation phase transition in connectivity. Consequently, the critical percolation properties of spreading processes on temporal networks can be estimated by a set of known exponents characterising the directed percolation universality class. This result is robust across a diverse set of temporal network models with different temporal and topological heterogeneities, while by using our methodology we uncover similar reachability phase transitions in real temporal networks too. These findings open up an avenue to apply theory, concepts and methodology from the well-developed directed percolation literature to temporal networks.

Critical behavior of nonequilibrium depinning transitions for vortices driven by current and vortex density

We study the critical dynamics of vortices associated with dynamic disordering near the depinning transitions driven by dc force (dc current I) and vortex density (magnetic field B). Independent of the driving parameters, I and B, we observe the critical behavior of the depinning transitions, not only on the moving side, but also on the pinned side of the transition, which is the first convincing verification of the theoretical prediction. Relaxation times, tau (I) and tau (B), to reach either the moving or pinned state, plotted against I and B, respectively, exhibit a power-law divergence at the depinning thresholds. The critical exponents of both transitions are, within errors, identical to each other, which are in agreement with the values expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. With an increase in B under constant I, the depinning transition at low B is replaced by the repinning transition at high B in the peak-effect regime. We find a trend that the critical exponents in the peak-effect regime are slightly smaller than those in the low-B regime and the theoretical one, which is attributed to the slight difference in the depinning mechanism in the peak-effect regime.

Phase Transition to Turbulence in Spatially Extended Shear Flows.

Directed percolation (DP) has recently emerged as a possible solution to the century old puzzle surrounding the transition to turbulence. Multiple model studies reported DP exponents, however, experimental evidence is limited since the largest possible observation times are orders of magnitude shorter than the flows' characteristic timescales. An exception is cylindrical Couette flow where the limit is not temporal, but rather the realizable system size. We present experiments in a Couette setup of unprecedented azimuthal and axial aspect ratios. Approaching the critical point to within less than 0.1% we determine five critical exponents, all of which are in excellent agreement with the 2+1D DP universality class. The complex dynamics encountered at the onset of turbulence can hence be fully rationalized within the framework of statistical mechanics.

Quantum and Classical Temporal Correlations in (1+1)D Quantum Cellular Automata.

We employ (1+1)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the "present" and the "past." We show how the uniquely quantum part of these correlations-the coherence-can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.

A random-walk-based epidemiological model

Random walkers on a two-dimensional square lattice are used to explore the spatio-temporal growth of an epidemic. We have found that a simple random-walk system generates non-trivial dynamics compared with traditional well-mixed models. Phase diagrams characterizing the long-term behaviors of the epidemics are calculated numerically. The functional dependence of the basic reproductive number R_{0} on the model’s defining parameters reveals the role of spatial fluctuations and leads to a novel expression for R_{0}. Special attention is given to simulations of inter-regional transmission of the contagion. The scaling of the epidemic with respect to space and time scales is studied in detail in the critical region, which is shown to be compatible with the directed-percolation universality class.