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arXiv:cond-mat/0406598

Hyperscaling relations in the bosonic pair contact process with diffusion

Matthias Paeßens

Published Jun 24, 2004• 0 citations
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Abstract

A hyperscaling relation for the critical exponents of absorbing phase transitions is tested in the bosonic pair contact process with diffusion. To this end spreading is considered, i.e. the time evolution out of an initial seed. It is shown that like in the case of spatial homogeneous initial conditions the autocorrelation function exhibits a phase transition at the critical point of the first moment. Some of the critical exponents can be determined exactly which is an unusual property of an absorbing phase transition and provides a possibility to test the hyperscaling relation. In the case of the bosonic pair contact process with diffusion three sets of exponents can be considered referring to the number of particles, number of pairs and number of active sites. It is argued that in special cases it is generally impossible to produce adequate data numerically.

DOI
10.48550/arxiv.cond-mat/0406598

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A hyperscaling relation for the critical exponents of absorbing phase transitions is tested in the bosonic pair contact process with diffusion. To this end spreading is considered, i.e. the time evolution out of an initial seed. It is shown that like in the case of spatial homogeneous initial conditions the autocorrelation function exhibits a phase transition at the critical point of the first moment. Some of the critical exponents can be determined exactly which is an unusual property of an absorbing phase transition and provides a possibility to test the hyperscaling relation. In the case of the bosonic pair contact process with diffusion three sets of exponents can be considered referring to the number of particles, number of pairs and number of active sites. It is argued that in special cases it is generally impossible to produce adequate data numerically.
The variance of the local density of the pair contact process with diffusion is investigated in a bosonic description. At the critical point of the absorbing phase transition (where the average particle number remains constant) it is shown that for a lattice dimension d > 2 the variance exhibits a phase transition: for high enough diffusion constants, it asymptotically approaches a finite value, while for low diffusion constants the variance diverges exponentially in time. This behaviour also appears in the density correlation function, implying that the correlation time is negative. Yet one has dynamical scaling with a dynamical exponent calculated to be z = 2.
The variance of the local density of the pair contact process with diffusion is investigated in a bosonic description. At the critical point of the absorbing phase transition (where the average particle number remains constant) it is shown that for a lattice dimension d > 2 the variance exhibits a phase transition: for high enough diffusion constants, it asymptotically approaches a finite value, while for low diffusion constants the variance diverges exponentially in time. This behaviour also appears in the density correlation function, implying that the correlation time is negative. Yet one has dynamical scaling with a dynamical exponent calculated to be z = 2.
Recently it was suggested that a pair contact process with diffusion (PCPD) might represent an independent new universality class different from the directed percolation (DP) and the parity conservation (PC) class. The dynamics in the PCPD are usually controlled by two independent parameters. The critical exponents for the PCPD are known to have different values for varying values of the two independent parameters. However, once the diffusion and annihilation (or coagulation) rate in the PCPD is tuned in a way that the process without offspring production is exactly solvable, a well-defined set of the exponents for the PCPD is obtained. Then dynamics are controlled by only one independent parameter. The obtained critical exponents are different than those of DP and PC. The critical exponents satisfy the generalized hyperscaling relation within numerical errors.
One-dimensional systems with an infinite number of absorbing states exhibit a phase transition that is not fully understood yet. Their static critical exponents are universal and belong in the Reggeon field theory (or directed percolation) universality class. However, exponents associated with the spreading of a localized seed appear to be nonuniversal depending on the nature of the initial condition. We investigate this problem by integrating numerically a non-Markovian Langevin equation proposed recently to describe such phase transitions. We find that the static critical exponents are universal, as expected. On the other hand, the Langevin equation reproduces the nonuniversal behavior observed in microscopic models for exponents associated with the spreading of an initially localized seed and satisfies the generalized hyperscaling relation proposed for those systems.
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.