Mean-Field Cluster Approximation Scheme for the Pair-Creation Model with Absorbing Phase Transition
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STATISTICAL
Mean-Field Cluster Approximation Scheme for the Pair-Creation Model with Absorbing Phase Transition Anderson A. Ferreira1 Received: 10 June 2020 / Accepted: 7 September 2020 / Published online: 17 September 2020 © Sociedade Brasileira de F´ısica 2020
Abstract We study the non-equilibrium phase transitions in the one-dimensional pair-creation model (2A → 3A, A → 0) using the n-site approximation scheme. We find the phase diagram in the space of the parameters (γ , D), where γ is the particle decay probability and D is the diffusion probability. Through data (1 ≤ n ≤ 18), we show that in the limit n → ∞ the model presents a continuous transition from active state to absorbing state for any value of 0 ≤ D ≤ 1. This result is in contrast with that of Fiore and de Oliveira (Phys. Rev. E 70, 046131, 2004) who found a discontinuous transition at high diffusion regime through conserved order parameter simulations. Keywords Mean-field cluster · Absorbing phase transition · Pair-creation model
1 Introduction In recent decades, there has been a great interest of statistical mechanics about one-dimensional systems that exhibit continuous [1–9] and discontinuous [10–16] phase transitions far from equilibrium. In particular, an argument created by Hinrichsen [17] assures that a single-component model (no longer interactions or additional symmetries) can not present a discontinuous phase transition from an active to an inactive (vacuum) state in one dimension. However, Fiore and de Oliveira [18] through the Monte Carlo simulation reported a tricritical point in high diffusion regime of the pair-creation model (PCM) (2A → 3A, A → 0). In order to analyze this apparent divergence, we review the phase diagram of the one-dimension PCM model with diffusion through the n-site approximation method [19] (mean field cluster approach). The localization of the tricritical point takes place where the particle density becomes discontinuous (gap) at the transition line (active to absorbing). Therefore, we can study the behavior of the gap ρn as a function of the cluster approximation n = 1, 2, . . . . inside the bidimensional Anderson A. Ferreira
[email protected] 1
Departamento de F´ısica, Universidade Federal de S˜ao Paulo, Diadema, S˜ao Paulo, Brazil
parameter space (γ , D). Here, γ is the decay probability in the reaction A → 0, and D is the diffusion probability to interchange a monomer A with near neighbor vacancy. We will show that in the single mean field approximation (n = 1), the gap ρ1 = 12 between active and absorbing state is constant. Moreover, in the pair site approximation (n = 2), the discontinuity ρ2 = 14 (γcD +D) is a monotonic function of D. However, in the limit n → ∞, we obtain (by extrapolation data) an unphysical tricritical point (γt = −0.03865 and Dt = 1.3494). This result suggests that the gap ρn→∞ → 0, and consequently the model, just exhibits a continuous phase transition. The rest of the paper is organized as follows: in Section 2, we describe the PCM model, whereas in Section 3 the n
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