Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2504.09082 · Symmetry

Revisiting the Contact Model with Diffusion Beyond the Conventional Methods

Roberto da Silva, Eliseu Venites Filho, Henrique A. Fernandes, Paulo Freitas Gomes

Published May 16, 2025• 1 citation
arXivHTMLPDF

Abstract

The contact process is a nonequilibrium Hamiltonian model that, even in one dimension, lacks an exact solution and has been extensively studied via Monte Carlo simulations, both in steady-state and time-dependent scenarios. Although the effects of particle mobility and diffusion on criticality have been preliminarily explored, they remain poorly understood in many aspects. In this work, we examine how the critical rate of the model varies with the probability of particle mobility. By analyzing different stochastic evolutions of the system, we employ two modern approaches: (1) Random Matrix Theory (RMT): By building on the success of RMT, particularly Wishart-like matrices, in studying statistical physics of systems with up-down symmetry via magnetization dynamics [R. da Silva, IJMPC 2022], we demonstrate its applicability to models with an absorbing state; (2) Optimized Temporal Power Laws: By using short-time dynamics, we optimize power laws derived from ensemble-averaged evolutions of the system. Both methods consistently reveal that the critical rate decays with mobility according to a simple Belehradek function. Additionally, a straightforward mean-field analysis supports the decay of the critical parameter with mobility, although it predicts a simpler linear dependence. We also demonstrate that the more sophisticated pair approximation mean-field model developed by ben-Avraham and Köhler aligns closely with the Belehradek function, precisely matching our lattice simulation results.

DOI
10.3390/sym17050774

Related papers

Coupled two-species model for the pair contact process with diffusion

Shengfeng Deng, Wei Li, Uwe Claus Täuber

The contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B, B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A, A→A+B, A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension d_{c}=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.

arXiv:2007.09496•Physical review. E•Oct 22, 2020•5 citations

Critical decay exponent of the pair contact process with diffusion

Su‐Chan Park

We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent δ. To obtain an accurate estimate of δ, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 62, 469 (2013)KPSJAS0374-488410.3938/jkps.62.469]. For small diffusion rate (d≤0.5), the leading corrections-to-scaling term is found to be ∼t^{-0.15}, whereas for large diffusion rate (d=0.95) it is found to be ∼t^{-0.5}. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent δ is 0.173(3) irrespective of d. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At d=0, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent ϕ=2.6(1). We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At d=1, the crossover from the mean field PCPD to the PCPD is described by ϕ=2 which is argued to be exact.

arXiv:1202.6024•Physical Review E•Nov 11, 2014•14 citations

Static critical behavior in the inactive phase of the pair contact process

M C Marques, Mariana Aparecida de Jesus Castro Santos, José F. F. Mendes

Steady-state properties in the absorbing phase of the $1d$ pair contact process model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles ${\ensuremath{\rho}}_{1},$ approaches a stationary value that depends on the annihilation probability $(p);$ the deviation from its ``natural'' value at criticality ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}$ follows a power law: ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}\ensuremath{-}{\ensuremath{\rho}}_{1}\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}_{1}}$ for $p>{p}_{c}.$ Monte Carlo simulations yield ${\ensuremath{\beta}}_{1}=0.81.$ A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting ${\ensuremath{\beta}}_{1}=1.$ The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.

arXiv:cond-mat/0109024•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Dec 14, 2001•8 citations

A renormalisation group study of one-dimensional contact processes

Jef Hooyberghs

In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.

•Jan 1, 2002•0 citations

On the critical behavior of the 1d diffusive pair contact process

Géza Ódor

The phase transition of the one-dimensional, diffusive pair contact process (PCPD) is investigated by N cluster mean-field approximations and high precision simulations. The N=3,4 cluster approximations exhibit smooth transition line to absorbing state by varying the diffusion rate D with beta_2=2 mean-field order parameter exponent of the pair density. This contradicts with former N=2 results, where two different mean-field behavior was found along the transition line. Extensive dynamical simulations on L=10^5 lattices give estimates for the order parameter exponents of the particles for 0.05 <= D <= 0.7. These data can support former two distinct class findings. However the gap between low and high D exponents is narrower than estimated previously and the possibility for interpreting numerical data as a single class behavior with exponents alpha=0.21(1), beta=0.41(1) assuming logarithmic corrections is shown. Finite size scaling and cluster simulation results are also presented.

•Sep 12, 2002•0 citations

Infinite-randomness critical point in the two-dimensional disordered contact process

Thomas Vojta, Adam Farquhar, Jason Mast

We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to 10;{10} and system sizes up to 8000x8000 sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.

arXiv:0810.1569•Physical Review E•Jan 12, 2009•122 citations
The contact process is a nonequilibrium Hamiltonian model that, even in one dimension, lacks an exact solution and has been extensively studied via Monte Carlo simulations, both in steady-state and time-dependent scenarios. Although the effects of particle mobility and diffusion on criticality have been preliminarily explored, they remain poorly understood in many aspects. In this work, we examine how the critical rate of the model varies with the probability of particle mobility. By analyzing different stochastic evolutions of the system, we employ two modern approaches: (1) Random Matrix Theory (RMT): By building on the success of RMT, particularly Wishart-like matrices, in studying statistical physics of systems with up-down symmetry via magnetization dynamics [R. da Silva, IJMPC 2022], we demonstrate its applicability to models with an absorbing state; (2) Optimized Temporal Power Laws: By using short-time dynamics, we optimize power laws derived from ensemble-averaged evolutions of the system. Both methods consistently reveal that the critical rate decays with mobility according to a simple Belehradek function. Additionally, a straightforward mean-field analysis supports the decay of the critical parameter with mobility, although it predicts a simpler linear dependence. We also demonstrate that the more sophisticated pair approximation mean-field model developed by ben-Avraham and Köhler aligns closely with the Belehradek function, precisely matching our lattice simulation results.
The contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B, B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A, A→A+B, A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension d_{c}=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.
We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent δ. To obtain an accurate estimate of δ, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 62, 469 (2013)KPSJAS0374-488410.3938/jkps.62.469]. For small diffusion rate (d≤0.5), the leading corrections-to-scaling term is found to be ∼t^{-0.15}, whereas for large diffusion rate (d=0.95) it is found to be ∼t^{-0.5}. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent δ is 0.173(3) irrespective of d. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At d=0, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent ϕ=2.6(1). We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At d=1, the crossover from the mean field PCPD to the PCPD is described by ϕ=2 which is argued to be exact.
Steady-state properties in the absorbing phase of the 1d1d1d pair contact process model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles ${\ensuremath{\rho}}_{1},$ approaches a stationary value that depends on the annihilation probability (p);(p);(p); the deviation from its ``natural'' value at criticality ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}$ follows a power law: ${\ensuremath{\rho}}_{1}^{\mathrm{nat}}\ensuremath{-}{\ensuremath{\rho}}_{1}\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}_{1}}$ for p>pc.p>{p}_{c}.p>pc​. Monte Carlo simulations yield ${\ensuremath{\beta}}_{1}=0.81.$ A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting ${\ensuremath{\beta}}_{1}=1.$ The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.
In the theory of stochastic many particle systems one models phenomena of non-equilibrium statistical mechanics by means of a Markov process. The master equation, describing the time evolution of such a process, can formally be interpreted as a Schrodinger equation in imaginary time. In this approach, the (non-Hermitian) generator of the process plays the role of the Hamiltonian. This mathematical equivalence permits the successful application to stochastic systems of a number of exact, approximative or numerical methods from quantum mechanics.A particularly interesting type of stochastic systems is formed by the processes having a phase transition between an absorbing state and an active state. Examples of these systems can be found in physics, chemistry and even biology. The best known process of this kind is the contact process. In this thesis real-space renormalisation group techniques, which were originally developed for quantum mechanical systems, are applied to study stochastic models on a one-dimensional lattice, more precisely the critical behaviour and the universality of absorbing state phase transitions. First, the standard renormalisation group (SRG) is adapted to a stochastic context resulting in one of the few analytical techniques capable of producing approximations for systems that can not be solved exactly. The SRG is successfully applied to the contact process for which it characterises the stationary critical behaviour with a competitive accuracy. Next a generalisation of the contact process to a model with several absorbing states is studied by means of the density matrix renormalisation group (DMRG) algorithm. The results for the cases with one and two absorbing states are consistent with what was known from simulations done by an other author. For the models with more than two absorbing states new results were found, which are supported by analytical considerations in a particular limit. Finally the experience of the previous two projects is used in the investigation of the quenched random contact process. In this model the local interactions on a lattice site are time-invariant stochastic variables, independently drawn from a given distribution with a certain variance, which gives a measure for the disorder. Very little was known about the stationary behaviour of this model since its extremely slow dynamics makes simulations inefficient. The renormalisation techniques in this thesis, suffer less from this problem. First, for the case of small disorder the SRG indicates that the critical behaviour is changed by the presence of disorder and the cross-over exponent is consistent with the Harris criterion. For intermediate disorder a DMRG study reveals that the critical exponents are continuously varying with the amount of disorder. Finally, an SRG calculation, suitably adapted to strong disorder, shows that in this case the phase transition is characterised by a fixed point at infinite randomness. In this limit the renormalisation is expected to be exact. It is the first time exact critical exponents are found for a phase transition out of an absorbing state. Moreover, the exponents are the same as those previously found in a class of disordered quantum chains, suggesting a new kind of universality for strongly disordered systems.
The phase transition of the one-dimensional, diffusive pair contact process (PCPD) is investigated by N cluster mean-field approximations and high precision simulations. The N=3,4 cluster approximations exhibit smooth transition line to absorbing state by varying the diffusion rate D with beta_2=2 mean-field order parameter exponent of the pair density. This contradicts with former N=2 results, where two different mean-field behavior was found along the transition line. Extensive dynamical simulations on L=10^5 lattices give estimates for the order parameter exponents of the particles for 0.05 <= D <= 0.7. These data can support former two distinct class findings. However the gap between low and high D exponents is narrower than estimated previously and the possibility for interpreting numerical data as a single class behavior with exponents alpha=0.21(1), beta=0.41(1) assuming logarithmic corrections is shown. Finite size scaling and cluster simulation results are also presented.
We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to 10;{10} and system sizes up to 8000x8000 sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.