We introduce a family of classical stochastic processes describing diffusive particles undergoing branching and long-range annihilation in the presence of a parity constraint. The probability for a pair-annihilation event decays as a power-law in the distance between particles, with a tunable exponent. Such long-range processes arise naturally in various classical settings, and increasingly play a role in the study of quantum dynamics: for example, quantum state preparation or error correction processes aim to prepare ordered ground states, which requires removing point-like excitations in pairs via non-local feedback operations conditioned on a global set of measurement outcomes. We analytically and numerically describe features of absorbing phases and phase transitions in this family of classical models as pairwise annihilation is performed at larger and larger distances. Notably, we find that the two canonical absorbing-state universality classes -- directed-percolation and parity-conserving -- are endpoints of a new line of universality classes with continuously varying critical exponents. These new classes follow from an interplay of long-range interactions and symmetries: decreasing the long-range exponent effectively weakens the local effects of global symmetry constraints. This competition significantly broadens known non-equilibrium phases and phase transitions.
A one dimensional non-equilibrium stochastic model is proposed where each site of the lattice is occupied by a particle, which may be of type A or B. The time evolution of the model occurs through three processes: autocatalytic generation of A and B particles and spontaneous conversion A to B. The two-parameter phase diagram of the model is obtained in one- and two-site mean field approximations, as well as through numerical simulations and exact solution of finite systems extrapolated to the thermodynamic limit. A continuous line of transitions between an active and an absorbing phase is foun…
We study the two-species diffusion-annihilation process, $A+B\rightarrow$ Ø, on the fully-connected lattice. Probability distributions for the number of particles and the reaction time are obtained for a finite-size system using a master equation approach. Mean values and variances are deduced from generating functions. When the reaction is far from complete, i.e., for a large number of particles of each species, mean-field theory is exact and the fluctuations are Gaussian. In the scaling limit the reaction time displays extreme-value statistics in the vicinity of the absorbing states. A gener…
Recently one has stated that temporal disorder constitutes a relevant perturbation in absorbing phase transitions for all dimensions. However, its effect for systems other than the standard contact process (CP), its competition with other ingredients (e.g. particle diffusion) and other kinds of disorder (besides the standard types) are unknown. In order to shed some light in the above mentioned points, we investigate a variant of the usual CP, namely triplet annihilation model (TAM), in which the competition between triplet annihilation and single particle diffusion leads to an unusual phase d…
We study in further detail particle models displaying a boundary-induced absorbing state phase transition [Phys. Rev. E. {\bf 65}, 046104 (2002) and Phys. Rev. Lett. {\bf 100}, 165701 (2008)] . These are one-dimensional systems consisting of a single site (the boundary) where creation and annihilation of particles occur and a bulk where particles move diffusively. We study different versions of these models, and confirm that, except for one exactly solvable bosonic variant exhibiting a discontinuous transition and trivial exponents, all the others display non-trivial behavior, with critical ex…
Contact processes (CP's) with particle creation requiring a minimal neighborhood (restrictive or threshold CP's) present a novel sort of discontinuous absorbing transitions, that revealed itself robust under the inclusion of different ingredients, such as distinct lattice topologies, particle annihilations and diffusion. Here, we tackle on the influence of competition between restrictive and standard dynamics (that describes the usual CP and a continuous DP transition is presented). Systems have been studied via mean-field theory (MFT) and numerical simulations. Results show partial contrast b…
We consider an extension of classical stochastic reaction-diffusion (RD) dynamics to open quantum systems. We study a class of models of hard core particles on a one-dimensional lattice whose dynamics is generated by a quantum master operator and where particle hopping is coherent while reactions, such as pair annihilation or pair coalescence, are dissipative. These are quantum open generalisations of the $A+A \to \varnothing$ and $A+A \to A$ classical RD models. We characterise the relaxation of the state towards the stationary regime via a decomposition of the system Hilbert space into trans…
We study the absorbing phase transitions in coupled directed percolation (DP) processes with $N$-species particles in one dimension. The interspecies coupling is linear, bidirectional, and excitatory. We find that the presence of a spontaneous annihilation process $A\to 0$ is essential in stabilizing the absorbing phase (vacuum). In the coupled contact processes, the vacuum is stable and the system exhibits DP type transitions, regardless of the coupling strength, for all $N$. However, in the coupled branching annihilation random walks with one offspring (BAW), where particle annihilations occ…
Motivated by recent findings, we discuss the existence of a direct and robust mechanism providing discontinuous absorbing transitions in short range systems with single species, with no extra symmetries or conservation laws. We consider variants of the contact process, in which at least two adjacent particles (instead of one, as commonly assumed) are required to create a new species. Many interaction rules are analyzed, including distinct cluster annihilations and a modified version of the original pair contact process (PCP). Through detailed time dependent numerical simulations we find that f…
Reaction-diffusion systems which include processes of the form A+A->A or A+A->0 are characterised by the appearance of `imaginary' multiplicative noise terms in an effective Langevin-type description. However, if `real' as well as `imaginary' noise is present, then competition between the two could potentially lead to novel behaviour. We thus investigate the asymptotic properties of the following two `mixed noise' reaction-diffusion systems. The first is a combination of the annihilation and scattering processes 2A->0, 2A->2B, 2B->2A, and 2B->0. We demonstrate (to all orders in perturbation th…
Using inverse Ising inference we show that the absorbing states of the one-dimensional Manna model can be described by an equilibrium model with an emergent interaction displaying short-ranged repulsion and long-ranged attraction. As the model approaches the critical point the interaction becomes purely repulsive, decaying as $r^{-α}$ and we conjecture the exact value $α=1/2$, suggesting density fluctuations decay as $r^{-3/2}$. We present a simple Gaussian field theory for the long distance behaviour of critical absorbing states and discuss implications for the Manna universality class.
We study a contact process with creation at first- and second-neighbor sites and inhibition at first neighbors, in the form of an annihilation rate that increases with the number of occupied first neighbors. Mean-field theory predicts three phases: inactive (absorbing), active symmetric, and active asymmetric, the latter exhibiting distinct sublattice densities on a bipartite lattice. These phases are separated by continuous transitions; the phase diagram is reentrant. Monte Carlo simulations in two dimensions verify these predictions qualitatively, except for a first-neighbor creation rate of…
It has been proposed (Phys. Rev. E {\bf 71}, 026121 (2005)) that unlike the short range contact process, a long-range counterpart may lead to the existence a discontinuous phase transition in one dimension. Aiming at exploring such link, here we investigate thoroughly a family of long-range contact processes. They are introduced through the transition rate $1+a\ell^{-σ}$, where $\ell$ is the length of inactive islands surrounding particles. In the former approach we reconsider the original model (called $σ-$contact process), by considering distinct mechanisms of weakening the long-range intera…
One of the most impressive features of continuous phase transitions is the concept of universality, that allows to group the great variety of different critical phenomena into a small number of universality classes. All systems belonging to a given universality class have the same critical exponents, and certain scaling functions become identical near the critical point. It is the aim of this work to demonstrate the usefulness of universal scaling functions for the analysis of non-equilibrium phase transitions. In order to limit the coverage of this article, we focus on a particular class of n…
We study phase transitions in a long-range one-dimensional cellular automaton with two symmetric absorbing states. It includes and extends several other models, like the Ising and Domany-Kinzel ones. It is characterized by a competing ferromagnetic linear coupling and an antiferromagnetic nonlinear one. Despite its simplicity, this model exhibits an extremely rich phase diagram. We present numerical results and mean-field approximations.
The paper addresses the single-file diffusion in the presence of an absorbing boundary. The emphasis is on an interplay between the hard-core interparticle interaction and the absorption process. The resulting dynamics exhibits several qualitatively new features. First, starting with the exact probability density function for a given particle (a tracer), we study the long-time asymptotics of its moments. Both the mean position and the mean square displacement are controlled by dynamical exponents which depend on the initial order of the particle in the file. Secondly, conditioning on non-absor…
The Monte Carlo with Absorbing Markov Chains (MCAMC) method is introduced. This method is a generalization of the rejection-free method known as the $n$-fold way. The MCAMC algorithm is applied to the study of the very low-temperature properties of the lifetime of the metastable state of Ising ferromagnets. This is done both for square-lattice and cubic-lattice nearest-neighbor models. Comparison is made with exact low-temperature predictions, in particular the low-temperature predictions that the metastable lifetime is discontinuous at particular values of the field. This discontinuity for th…
Many-body unitary dynamics interspersed with repeated measurements display a rich phenomenology hallmarked by measurement-induced phase transitions. Employing feedback-control operations that steer the dynamics toward an absorbing state, we study the entanglement entropy behavior at the absorbing state phase transition. For short-range control operations, we observe a transition between phases with distinct sub-extensive scalings of entanglement entropy. In contrast, the system undergoes a transition between volume-law and area-law phases for long-range feedback operations. The fluctuations of…
A conserved lattice gas with random neighbor hopping of active particles is introduced which exhibits a continuous phase transition from an active state to an absorbing non-active state. Since the randomness of the particle hopping breaks long range spatial correlations our model mimics the mean-field scaling behavior of the recently introduced new universality class of absorbing phase transitions with a conserved field. The critical exponent of the order parameter is derived within a simple approximation. The results are compared with those of simulations and field theoretical approaches.
In a recent article [arXiv:1108.5127] Park has shown that the four-state predator-prey model studied earlier in [J. Stat. Mech, L05001 (2011)] belongs to Directed Percolation (DP) universality class. It was claimed that predator density is not a reasonable order parameter, as there are many absorbing states; a suitably chosen order parameter shows DP critical behavior. In this article, we argue that the configuration that does not have any predator is the only dynamically accessible absorbing configuration, and the predator density too settles to DP critical exponents after a long transient.
We introduce a discrete-time quantum dynamics on a two-dimensional lattice that describes the evolution of a $1+1$-dimensional spin system. The underlying quantum map is constructed such that the reduced state at each time step is separable. We show that for long times this state becomes stationary and displays a continuous phase transition in the density of excited spins. This phenomenon can be understood through a connection to the so-called Domany-Kinzel automaton, which implements a classical non-equilibrium process that features a transition to an absorbing state. Near the transition dens…
The presence of a unique absorbing state in the finite state space of an epidemic process always poses a challenge in the definition of its epidemiological threshold. Without the elimination of the absorbing state of the SIS process we show that an exponentially distributed finite lifetime of the population results in a threshold like beahivour in the long time limit. We illustrate our findings with the SIS dynamics on the complete graph and the star.
The well-established universality classes of absorbing critical phenomena are directed percolation (DP) and directed Ising (DI) classes. Recently, the pair contact process with diffusion (PCPD) has been investigated extensively and claimed to exhibit a new type of critical phenomena distinct from both DP and DI classes. Noticing that the PCPD possesses a long-term memory effect, we introduce a generalized version of the PCPD (GPCPD) with a parameter controlling the memory effect. The GPCPD connects the DP fixed point to the PCPD point continuously. Monte Carlo simulations show that the GPCPD d…
We study the long-time behavior of conservative interacting particle systems in $Z$: the activated random walk model for reaction-diffusion systems and the stochastic sandpile. We prove that both systems undergo an absorbing-state phase transition.
We describe how to solve the problem of Taylor dispersion in the presence of absorbing boundaries using an exact stochastic formulation. In addition to providing a clear stochastic picture of Taylor dispersion, our method leads to closed-form expressions for all the moments of the convective displacement of the dispersing particles in terms of the transverse diffusion eigenmodes. We also find that the cumulants grow asymptotically linearly with time, ensuring a Gaussian distribution in the long-time limit. As a demonstration of the technique, the first two longitudinal cumulants (yielding resp…
We elucidate a long-standing puzzle about the non-equilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved non-diffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.
The directed bond percolation process is studied in the presence of com- pressible velocity fluctuations with long-range correlations. We discuss a construction of a field theoretic action and a way of obtaining its large scale properties using the perturbative renormalization group. The most interest- ing results for the frozen velocity limit are given.
We study the long-time behavior of stochastic models with an absorbing state, conditioned on survival. For a large class of processes, in which saturation prevents unlimited growth, statistical properties of the surviving sample attain time-independent limiting values. We may then define a quasi-stationary probability distribution as one in which the ratios p_n(t)/p_m(t) (for any pair of nonabsorbing states n and m), are time-independent. This is not a true stationary distribution, since the overall normalization decays as probability flows irreversibly to the absorbing state. We construct qua…
We calculate the exact asymptotic survival probability, Q, of a one-dimensional Brownian particle, initially located located at the point x in (-L,L), in the presence of two moving absorbing boundaries located at \pm(L+ct). The result is Q(y,λ) = \sum_{n=-\infty}^\infty (-1)^n \cosh(ny) \exp(-n^2λ), where y=cx/D, λ= cL/D and D is the diffusion constant of the particle. The results may be extended to the case where the absorbing boundaries have different speeds. As an application, we compute the asymptotic survival probability for the trapping reaction A + B -> B, for evanescent traps with a lo…
Nonequilibrium phase transitions between an active and an absorbing state are found in models of populations, epidemics, autocatalysis, and chemical reactions on a surface. While absorbing-state phase transitions fall generically in the DP universality class, this does not preclude other universality classes, associated with a symmetry or conservation law. An interesting issue concerns the dynamic critical behavior of models with an infinite number of absorbing configurations or a long memory. Sandpile models, the principal example of self-organized criticality (SOC), also exhibit absorbing- s…
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 exp…
A simple example of a non-equilibrium system for which fluctuations are important is a system of particles which diffuse and may annihilate in pairs on contact. The renormalization group can be used to calculate the time dependence of the density of particles, and provides both an exact value for the exponent governing the decay of particles and an epsilon-expansion for the amplitude of this power law. When the diffusion is anomalous, as when the particles perform Levy flights, the critical dimension depends continuously on the control parameter for the Levy distribution. The epsilon-expansion…
A granular system confined in a quasi two-dimensional box that is vertically vibrated can transit to an absorbing state in which all particles bounce vertically in phase with the box, with no horizontal motion. In principle, this state can be reached for any density lower than the one corresponding to one complete monolayer, which is then the critical density. Below this critical value, the transition to the absorbing state is of first order, with long metastable periods, followed by rapid transitions driven by homogeneous nucleation. Molecular dynamics simulations and experiments show that th…
We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive numerical simulations for several velocity distributions. This leads us to the conjecture that all the continuous velocity distributions ϕ(v) which are symmetric, regular and such that ϕ(0) does not vanish, are attracted in the long time regime towards the same Gaussian distribution and thus belong to the same universality class. Moreover, it is found that the…
We consider a diffusion-limited reaction in case the reacting entities are not available simultaneously. Due to the fact that the reaction takes place after a spatiotemporal accumulation of reactants, the underlying rate equation has to be modified by additional non-local terms. Owing to the delay effects a finite amount of reactants remains localized, preventing a further reaction and the asymptotic decay is terminated at a finite density. The resulting inhomogeneous non-zero stationary concentration is stable against long wave length fluctuations. Below a critical wave vector $k_c$ the syste…
It is known that diffusion provokes substantial changes in continuous absorbing phase transitions. Conversely, its effect on discontinuous transitions is much less understood. In order to shed light in this direction, we study the inclusion of diffusion in the simplest one-dimensional model with a discontinuous absorbing phase transition, namely the long-range contact process ($σ$-CP). Particles interact as in the usual CP, but the transition rate depends on the length $\ell$ of inactive sites according to $1 + a \ell^{-σ}$, where $a$ and $σ$ are control parameters. In the absence of diffusion…
Understanding and probing phase transitions in non-equilibrium systems is an ongoing challenge in physics. A particular instance are phase transitions that occur between a non-fluctuating absorbing phase, e.g., an extinct population, and one in which the relevant order parameter, such as the population density, assumes a finite value. Here we report the observation of signatures of such a non-equilibrium phase transition in an open driven quantum system. In our experiment rubidium atoms in a quasi one-dimensional cold disordered gas are laser-excited to Rydberg states under so-called facilitat…
The behavior of a heavy tagged intruder immersed in a bath of particles evolving under ballistic annihilation dynamics is investigated. The Fokker-Planck equation for this system is derived and the peculiarities of the corresponding diffusive behavior are worked out. In the long time limit, the intruder velocity distribution function approaches a Gaussian form, but with a different temperature from its bath counterpart. As a consequence of the continuous decay of particles in the bath, the mean squared displacement increases exponentially in the collision per particle time scale. Analytical re…
We determine how long a diffusing particle spends in a given spatial range before it dies at an absorbing boundary. In one dimension, for a particle that starts at $x_0$ and is absorbed at $x=0$, the average residence time in the range $[x,x+dx]$ is $T(x)=\frac{x}{D}\,dx$ for $x<x_0$ and $\frac{x_0}{D}\,dx$ for $x>x_0$, where $D$ is the diffusion coefficient. We extend our approach to biased diffusion, to a particle confined to a finite interval, and to general spatial dimensions. We use the generating function technique to derive parallel results for the average residence time of the one-dime…
It is widely assumed that there exists a simple transformation from the Itô interpretation to the one by Stratonovich and back for any stochastic differential equation of applied interest. While this transformation exists under suitable conditions, and transforms one interpretation into the other at the price of modifying the drift of the equation, it cannot be considered universal. We show that a class of stochastic differential equations, characterized by the presence of absorbing states and of interest in applications, does not admit such a transformation. In particular, formally applying t…
Universal interface fluctuations in absorbing-state phase transitions
RIKEN Pioneering Research Institute（理化学研究所先锋研究本部）；RIKEN Center for Biosystems Dynamics Research（理化学研究所生命动态研究中心）；The University of Tokyo（东京大学）；Institute for Physics of Intelligence, The University of Tokyo（东京大学智能物理研究所）；The Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Academia Sinica（中央研究院）；National Center for Theoretical Sciences（国家理论科学中心）；Mechanobiology Institute, National University of Singapore（新加坡国立大学生物力学研究所）；Universal Biology Institute, The University of Tokyo（东京大学泛生物研究所）
We consider a modified Nagel-Schreckenberg (NS) model in which drivers do not decelerate if their speed is smaller than the headway (number of empty sites to the car ahead). (In the original NS model, such a reduction in speed occurs with probability $p$, independent of the headway, as long as the current speed is greater than zero.) In the modified model the free-flow state (with all vehicles traveling at the maximum speed, $v_{max}$) is {\it absorbing} for densities $ρ$ smaller than a critical value $ρ_c = 1/(v_{max} + 2)$. The phase diagram in the $ρ- p$ plane is reentrant: for densities in…
