Directed Percolation Research Articles

Anomalous Directed Percolation on a Dynamic Network Using Rydberg Facilitation.

The facilitation of Rydberg excitations in a gas of atoms provides an ideal model system to study epidemic evolution on (dynamic) networks and self-organization of complex systems to the critical point of a nonequilibrium phase transition. Using MonteCarlo simulations and a machine learning algorithm we show that the universality class of this phase transition can be tuned but is robust against decay inherent to the self-organization process. The classes include directed percolation (DP), the most common class in short-range spreading models, and mean-field (MF) behavior, but also different types of anomalous directed percolation (ADP), characterized by rare long-range excitation processes. In a frozen gas, ground state atoms that can facilitate each other form a static network, for which we predict DP universality. With atomic motion the network becomes dynamic by long-range (Lévy-flight type) excitations. This leads to continuously varying critical exponents, varying smoothly between DP and MF values, corresponding to the ADP universality class. These findings also explain the recently observed critical exponent of Rydberg facilitation in an ultracold gas experiment [Helmrich etal., Nature (London) 577, 481 (2020)NATUAS0028-083610.1038/s41586-019-1908-6], which was in between DP and MF values.

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.

Supervised and unsupervised learning of (1+1) -dimensional even-offspring branching annihilating random walks

Machine learning (ML) of phase transitions (PTs) has gradually become an effective approach that enables us to explore the nature of various PTs more promptly in equilibrium and nonequilibrium systems. Unlike equilibrium systems, non-equilibrium systems display more complicated and diverse features because of the extra dimension of time, which is not readily tractable, both theoretically and numerically. The combination of ML and most renowned nonequilibrium model, directed percolation (DP), led to some significant findings. In this study, ML is applied to (1+1) -d, even offspring branching annihilating random walks (BAW), whose universality class is not DP-like. The supervised learning of (1+1) -d BAW via convolutional neural networks (CNN) results in a more accurate prediction of the critical point than the Monte Carlo (MC) simulation for the same system sizes. The dynamic exponent z and spatial correlation length correlation exponent ν⊥ were also measured and found to be consistent with their respective theoretical values. Furthermore, the unsupervised learning of (1+1) -d BAW via an autoencoder (AE) gives rise to a transition point, which is the same as the critical point. The latent layer of AE, through a single neuron, can be regarded as the order parameter of the system being properly re-scaled. Therefore, we believe that ML has exciting application prospects in reaction-diffusion systems such as BAW and DP.

Many universality classes in an interface model restricted to non-negative heights.

We present a simple one-dimensional stochastic model with three control parameters and a surprisingly rich zoo of phase transitions. At each (discrete) site x and time t, an integer n(x,t) satisfies a linear interface equation with added random noise. Depending on the control parameters, this noise may or may not satisfy the detailed balance condition, so that the growing interfaces are in the Edwards-Wilkinson or in the Kardar-Parisi-Zhang universality class. In addition, there is also a constraint n(x,t)≥0. Points x where n>0 on one side and n=0 on the other are called "fronts." These fronts can be "pushed" or "pulled," depending on the control parameters. For pulled fronts, the lateral spreading is in the directed percolation (DP) universality class, while it is in a different universality class for pushed fronts, and another universality class in between. In the DP case, the activity at each active site can in general be arbitrarily large, in contrast to previous realizations of DP. Finally, we find two different types of transitions when the interface detaches from the line n=0 (with 〈n(x,t)〉→const on one side, and →∞ on the other), again with new universality classes. We also discuss a mapping of this model to the avalanche propagation in a directed Oslo rice pile model in specially prepared backgrounds.

Machine learning of pair-contact process with diffusion

The pair-contact process with diffusion (PCPD), a generalized model of the ordinary pair-contact process (PCP) without diffusion, exhibits a continuous absorbing phase transition. Unlike the PCP, whose nature of phase transition is clearly classified into the directed percolation (DP) universality class, the model of PCPD has been controversially discussed since its infancy. To our best knowledge, there is so far no consensus on whether the phase transition of the PCPD falls into the unknown university classes or else conveys a new kind of non-equilibrium phase transition. In this paper, both unsupervised and supervised learning are employed to study the PCPD with scrutiny. Firstly, two unsupervised learning methods, principal component analysis (PCA) and autoencoder, are taken. Our results show that both methods can cluster the original configurations of the model and provide reasonable estimates of thresholds. Therefore, no matter whether the non-equilibrium lattice model is a random process of unitary (for instance the DP) or binary (for instance the PCP), or whether it contains the diffusion motion of particles, unsupervised learning can capture the essential, hidden information. Beyond that, supervised learning is also applied to learning the PCPD at different diffusion rates. We proposed a more accurate numerical method to determine the spatial correlation exponent nu _{perp }, which, to a large degree, avoids the uncertainty of data collapses through naked eyes.

Transfer learning of phase transitions in percolation and directed percolation.

The latest advances of statistical physics have shown remarkable performance of machine learning in identifying phase transitions. In this paper, we apply domain adversarial neural network (DANN) based on transfer learning to studying nonequilibrium and equilibrium phase transition models, which are percolation model and directed percolation (DP) model, respectively. With the DANN, only a small fraction of input configurations (two-dimensional images) needs to be labeled, which is automatically chosen, to capture the critical point. To learn the DP model, the method is refined by an iterative procedure in determining the critical point, which is a prerequisite for the data collapse in calculating the critical exponent ν_{⊥}. We then apply the DANN to a two-dimensional site percolation with configurations filtered to include only the largest cluster which may contain the information related to the order parameter. The DANN learning of both models yields reliable results which are comparable to the ones from Monte Carlo simulations. Our study also shows that the DANN can achieve quite high accuracy at much lower cost, compared to the supervised learning.

The elastic and directed percolation backbone

We argue that the elastic backbone (EB) (union of shortest paths) on a cylindrical system, studied by Sampaio Filho et al [2018 Phys. Rev. Lett. 120 175701], is in fact the backbone of two-dimensional directed percolation (DP). We simulate the EB on the same system as considered by these authors, and also study the DP backbone directly using an algorithm that allows backbones to be generated in a completely periodic manner. We find that both the EB in the bulk and the DP backbone have a fractal dimension of d b = d B,DP = 1.681 02(15) at the identical critical point p c,DP ≈ 0.705 485 22. We also measure the fractal dimension at the edge of the EB system and for the full DP clusters, and find d e = d DP = 1.840 54(4). We argue that those two fractal dimensions follow from the DP exponents as d B,DP = 2 − 2β/ν ∥ = 1.681 072(12) and d DP = 2 − β/ν ∥ = 1.840 536(6). Our fractal dimensions differ from the value 1.750(3) found by Sampaio Filho et al.

Alternative method for measuring characteristic lengths in absorbing phase transitions.

We applied an alternative method for measuring characteristic lengths reported recently by one of us [J. M. Kim, J. Stat. Mech. (2021) 03321310.1088/1742-5468/abe599] to the models in the Manna universality class, i.e., the stochastic Manna sandpile and conserved lattice gas models in various dimensions. The universality of the Manna model has been under long debate particularly in one dimension since the work of M. Basu etal. [Phys. Rev. Lett. 109, 015702 (2012)10.1103/PhysRevLett.109.015702], who claimed that the Manna model belongs to the directed percolation (DP) universality class and that the independent Manna universality class does not exist. We carried out Monte Carlo simulations for the stochastic Manna sandpile model in one, two, and three dimensions and the conserved lattice gas model in three dimensions, using both the natural initial states (NISs) and uniform initial states (UISs). In two and three dimensions, the results for R(t), defined by R(t)=L[〈ρ_{a}^{2}〉/〈ρ_{a}〉^{2}-1]^{1/d},L and ρ_{a} being, respectively, the system size and activity density, yielded consistent results for the two initial states. R(t) is proportional to the correlation length following R(t)∼t^{1/z} at the critical point. In one dimension, the data of R(t) for the Manna model using NISs yielded anomalous behavior, suggesting that NISs require much longer prerun time steps to homogenize the distribution of particles and larger systems to eliminate the finite-size effect than those employed in the literature. On the other hand, data from UISs yielded a power-law behavior, and the estimated critical exponents differed from the values in the DP class.

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.

Negotiation problem

We propose and solve a negotiation model of multiple players facing many alternative solutions. The model can be generalized to many relevant circumstances where stakeholders’ interests partially overlap and partially oppose. We also show that the model can be mapped into the well-known directed percolation and directed polymers problems. Moreover, many statistical mechanics tools, such as the Replica method, can be fruitfully employed. Studying our negotiation model can enlighten the links between social–economic phenomena and traditional statistical mechanics and help to develop new perspectives and tools in the fertile interdisciplinary field.

Effects of the velocity on the reversible-irreversible transition in a periodically sheared vortex system

A reversible-irreversible transition (RIT) is studied using a periodically-driven vortex system in an amorphous film with random pinning that causes local shear, as a function of shear amplitude d. The relaxation time to reach the steady state exhibits a power-law divergence at a threshold value dc with critical exponents in agreement with the values predicted for an absorbing phase transition in the two-dimensional (2D) directed-percolation (DP) universality class. In our previous work, the experiment was conducted at relatively high frequency f, giving rise to a large mean vortex velocity v. Here we use lower f to study the effects of reduced v and increased dynamic pinning on the RIT. The results show that the critical behavior of RIT stays essentially unchanged, while we find a trend for dc to increase with decreasing v. We will propose a possible model to qualitatively explain this unexpected result.

Supervised and unsupervised learning of directed percolation.

Machine learning (ML) has been well applied to studying equilibrium phase transition models by accurately predicating critical thresholds and some critical exponents. Difficulty will be raised, however, for integrating ML into nonequilibrium phase transitions. The extra dimension in a given nonequilibrium system, namely time, can greatly slow down the procedure toward the steady state. In this paper we find that by using some simple techniques of ML, non-steady-state configurations of directed percolation (DP) suffice to capture its essential critical behaviors in both (1+1) and (2+1) dimensions. With the supervised learning method, the framework of our binary classification neural networks can identify the phase transition threshold, as well as the spatial and temporal correlation exponents. The characteristic time t_{c}, specifying the transition from active phases to absorbing ones, is also a major product of the learning. Moreover, we employ the convolutional autoencoder, an unsupervised learning technique, to extract dimensionality reduction representations and cluster configurations of (1+1) bond DP. It is quite appealing that such a method can yield a reasonable estimation of the critical point.

Absorbing phase transition with a continuously varying exponent in a quantum contact process: A neural network approach

Phase transitions in dissipative quantum systems are intriguing because they are induced by the interplay between coherent quantum and incoherent classical fluctuations. Here, we investigate the crossover from a quantum to a classical absorbing phase transition arising in the quantum contact process (QCP). The Lindblad equation contains two parameters, $\omega$ and $\kappa$, which adjust the contributions of the quantum and classical effects, respectively. We find that in one dimension when the QCP starts from a homogeneous state with all active sites, there exists a critical line in the region $0 \le \kappa < \kappa_*$ along which the exponent $\alpha$ (which is associated with the density of active sites) decreases continuously from a quantum to the classical directed percolation (DP) value. This behavior suggests that the quantum coherent effect remains to some extent near $\kappa=0$. However, when the QCP in one dimension starts from a heterogeneous state with all inactive sites except for one active site, all the critical exponents have the classical DP values for $\kappa \ge 0$. In two dimensions, anomalous crossover behavior does not occur, and classical DP behavior appears in the entire region of $\kappa \ge 0$ regardless of the initial configuration. Neural network machine learning is used to identify the critical line and determine the correlation length exponent. Numerical simulations using the quantum jump Monte Carlo technique and tensor network method are performed to determine all the other critical exponents of the QCP.

Transitional Channel Flow: A Minimal Stochastic Model.

In line with Pomeau’s conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes observed in channel flow before its return to laminar flow. Numerical simulations show that a regime with bands obliquely drifting in two stream-wise symmetrical directions bifurcates into an asymmetrical regime, before ultimately decaying to laminar flow. The model is expressed in terms of a probabilistic cellular automaton of evolving von Neumann neighborhoods with probabilities educed from a close examination of simulation results. It implements band propagation and the two main local processes: longitudinal splitting involving bands with the same orientation, and transversal splitting giving birth to a daughter band with an orientation opposite to that of its mother. The ultimate decay stage observed to display one-dimensional DP properties in a two-dimensional geometry is interpreted as resulting from the irrelevance of lateral spreading in the single-orientation regime. The model also reproduces the bifurcation restoring the symmetry upon variation of the probability attached to transversal splitting, which opens the way to a study of the critical properties of that bifurcation, in analogy with thermodynamic phase transitions.

CONTACT PROCESS ON FRACTAL CLUSTERS SIMULATED BY GENERALIZED DIFFUSION-LIMITED AGGREGATION (g-DLA) MODEL

The spread of infectious disease, virus epidemic, fashion, religion and rumors is strongly affected by the nearest neighbor hence underlying morphologies of the colonies are crucial. Likewise, the morphology of naturally grown patterns ranges from fractal to compact with lacunarity. We analyze the contact process on the fractal clusters simulated by generalized Diffusion-limited Aggregation (g-DLA) model. In g-DLA model, randomly walking particle is added to the cluster with sticking probability [Formula: see text] depending on the local density of occupied sites in the neighborhood of radius [Formula: see text] from the center of active site. It takes values [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text]) for highly dense, moderately dense and sparsely occupied regions, respectively. The corresponding morphology varies from fractal to compact as [Formula: see text] varies from [Formula: see text] to [Formula: see text]. Interestingly, the contact process on the g-DLA clusters shows clear transition from active phase to absorbing phase and the exponent values fall between 1-d and 2-d in directed percolation (DP) universality class. The local persistence exponents at transition are studied and are found to be much smaller than that for 1-d and 2-d DP cases. We conjecture that infection in the fractal cluster does not easily reach far-flung or remote areas at the periphery of the cluster.

Critical phenomena in the presence of symmetric absorbing states: A microscopic spin model with tunable parameters.

The Langevin description of systems with two symmetric absorbing states yields a phase diagram with three different phases (disordered and active, ordered and active, absorbing) separated by critical lines belonging to three different universality classes (generalized voter, Ising, and directed percolation). In this paper we present a microscopic spin model with two symmetric absorbing states that has the property that the model parameters can be varied in a continuous way. Our results, obtained through extensive numerical simulations, indicate that all features of the Langevin description are encountered for our two-dimensional microscopic spin model. Thus the Ising and direction percolation lines merge into a generalized voter critical line at a point in parameter space that is not identical to the classical voter model. A vast range of different quantities are used to determine the universality classes of the order-disorder and absorbing phase transitions. The investigation of time-dependent quantities at a critical point belonging to the generalized voter universality class reveals a more complicated picture than previously discussed in the literature.

Physics of psychophysics: Large dynamic range in critical square lattices of spiking neurons

Psychophysics try to relate physical input magnitudes to psychological or neural correlates. Microscopic models to account for macroscopic psychophysical laws, in the sense of statistical physics, are an almost unexplored area. Here we examine a sensory epithelium composed of two connected square lattices of stochastic integrate-and-fire cells. With one square lattice we obtain a Stevens's law $\rho \propto h^m$ with Stevens's exponent $m = 0.254$ and a sigmoidal saturation, where $\rho$ is the neuronal network activity and $h$ is the input intensity (external field). We relate Stevens's power law exponent with the field critical exponent as $m = 1/\delta_h = \beta/\sigma$. We also show that this system pertains to the Directed Percolation (DP) universality class (or perhaps the Compact-DP class). With stacked two layers of square lattices, and a fraction of connectivity between the first and second layer, we obtain at the output layer $\rho_ 2 \propto h^{m_2}$, with $m_2 = 0.08 \approx m^2$, which corresponds to a huge dynamic range. This enhancement of the dynamic range only occurs when the layers are close to their critical point.

Subcritical Laminar–Turbulent Transition as Nonequilibrium Phase Transition in Two-Dimensional Kolmogorov Flow

We have confirmed numerically that a subcritical laminar–turbulent transition that belongs to the directed percolation (DP) universality class occurs in a purely two-dimensional (2D) simple Navier–...

Novel transition to fully absorbing state without long-range spatial order in directed percolation class

We study coupled Gauss maps in one dimension and observe a transition to band periodic state with 2 bands. This is a periodic state with period-2 in a coarse-grained sense. This state does not show any long-range order in space. We compute two different order parameters to quantify the transition a) Flipping rate F(t) which measures departures from period-2 and b) Persistence P(t) which quantifies the loss of memory of initial conditions. At the critical point, F(t) shows a power-law decay with exponent 0.158 which is close to 1-D directed percolation (DP) transition. The persistence exponent at the critical point is found to be 1.51 which matches with several models in 1-D DP class. We also study the finite-size scaling and off-critical scaling to estimate other exponents z and ν∥. We observe excellent scaling for both F(t) as well as P(t) and the exponents obtained are clearly in DP class. We believe that DP transition could be observed in systems where activity goes to zero even if the spatial profile could be inhomogeneous and lacking any long-range order.

Dynamic phase transition in the contact process with spatial disorder: Griffiths phase and complex persistence exponents.

We present a model which displays the Griffiths phase, i.e., algebraic decay of density with continuously varying exponents in the absorbing phase. In the active phase, the memory of initial conditions is lost with continuously varying complex exponents in this model. This is a one-dimensional model where a fraction r of sites obey rules leading to the directed percolation class and the rest evolve according to rules leading to the compact directed percolation class. For infection probability p≤p_{c}, the fraction of active sites ρ(t)=0 asymptotically. For p>p_{c},ρ(∞)>0. At p=p_{c},ρ(t), the survival probability from a single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence P_{l}(∞)>0 for p≤p_{c} and P_{l}(∞)=0 for p>p_{c}. For p≥p_{s}, local persistence P_{l}(t) decays as a power law with continuously varying exponents. The persistence exponent is clearly complex as p→1. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increase with p. We note that the underlying lattice or disorder does not have a self-similar structure.