Minimal mechanism leading to discontinuous phase transitions for short-range systems with absorbing states

Carlos E. Fiore

Phys. Rev. E 89, 022104 – Published 6 February 2014

Abstract

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. Through detailed time-dependent numerical simulations, we find that for our modified models, the phase transitions are of first order, hence contrasting with their corresponding usual formulations in the literature, which are of second order. By calculating the order-parameter distributions, the obtained bimodal shapes as well as the finite-scale analysis reinforce coexisting phases and thus a discontinuous transition. These findings strongly suggest that the above particle creation requirements constitute a minimum and fundamental mechanism determining the phase coexistence in short-range contact processes.

Received 17 July 2013
Revised 5 November 2013

DOI:https://doi.org/10.1103/PhysRevE.89.022104

Authors & Affiliations

Carlos E. Fiore

Instituto de Física, Universidade de São Paulo Caixa Postal 66318, 05315-970 São Paulo, Brazil

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Vol. 89, Iss. 2 — February 2014

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Images

Figure 1
Examples of transition rates in a square lattice ( $z = 4$ ). Models A, B, and C are defined by the (a)-(b), (a)-(c), and (a)-(d) interaction rules, respectively. In (d), the symbol $\times$ denotes a local configuration composed of $N$ nearest-neighbor occupied sites (with $N$ ranging from 0 to 4) and after the annihilation all $N + 1$ particles are extinct.
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Figure 2
Examples of transition rates for model D. Note that there is no particle creation if the number of pairs of particles $N_{p}$ is smaller than 2.
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Figure 3
For model A and distinct $α$ 's, we plot in (a) the time decay of the order parameter $ϕ$ evaluated over all and over only survived (inset) runs, respectively. In order to compare, we plot in (b) the time decay of the order parameter $ρ$ for distinct $α$ 's by considering the $N \geq 1$ creation with pair annihilation [23]. The black line in the middle curve has slope $θ = 0.4505 (10)$ . The upper inset in (b) shows the time decay of $ϕ$ (fraction of particles surrounded by at least $N = 1$ occupied sites) over all runs and the lower inset shows the time decay of $ρ$ measured over only survived runs. In (c) we plot the probability distribution $P_{ϕ}$ for distinct $L$ 's at $α_{L}$ , in which the peaks present the same height. The scaling plot of $α_{L}$ vs $L^{- 2}$ is shown in (d). In the inset, we show a log-log plot of steady order parameters $ϕ_{s t}$ , in the active $ϕ_{a c}$ and absorbing $ϕ_{a b}$ phases, vs $L$ .
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Figure 4
For model B and distinct $α$ 's, we plot in (a) the time decay of the order parameter $ϕ$ evaluated over all and over only survived (inset) runs, respectively. In order to compare, we plot in (b) the time decay of the order parameter $ρ$ for distinct $α$ 's by considering the $N \geq 1$ creation case with triplet annihilation. The black line in the middle curve has slope $θ = 0.4505 (10)$ . The inset in (b) shows the time decay of $ρ$ measured over only survived runs. In (c) we plot the probability distribution $P_{ϕ}$ for distinct $L$ 's at $α_{L}$ , in which the peaks present the same height. The scaling plot of $α_{L}$ vs $L^{- 2}$ is shown in (d). In the inset, we show a log-log plot of steady order parameters $ϕ_{s t}$ , in the active $ϕ_{a c}$ and absorbing $ϕ_{a b}$ phases, vs $L$ .
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Figure 5
For model C and distinct $α$ 's, we plot in (a) the time decay of the order parameter $ρ$ evaluated over all and over only survived (inset) runs, respectively. In order to compare, we plot in (b) the time decay of the order parameter $ρ$ for distinct $α$ 's by considering the $N \geq 1$ creation case, but with the same annihilation rule of model C. The black line in the middle curve has slope $θ = 0.4505 (10)$ . The inset in (b) shows the time decay of $ρ$ measured over only survived runs. In (c) we plot the probability distribution $P_{ρ}$ for distinct $L$ 's at $α_{L}$ , in which the peaks present the same height. The scaling plot of $α_{L}$ vs $L^{- 2}$ is shown in (d). In the inset, we show a log-log plot of steady order parameters $ρ_{s t}$ , in the active $ρ_{a c}$ and absorbing $ρ_{a b}$ phases, vs $L$ .
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Figure 6
For the model D and distinct $α$ 's, we plot in (a) the time decay of the order parameter $ϕ$ evaluated over all and over only survived (inset) runs, respectively. In order to compare, we plot in (b) the time decay of the order parameter $ϕ$ for distinct $α$ 's for the original PCP. The black line in the middle curve has slope $θ = 0.4505 (10)$ . The inset in (b) shows the time decay of $ϕ$ measured over only survived runs. In (c) we plot the probability distribution $P_{ϕ}$ for distinct $L$ 's at $α_{L}$ , in which the peaks present the same height. The scaling plot of $α_{L}$ vs $L^{- 2}$ is shown in (d). In the inset, we show a log-log plot of steady order parameters $ϕ_{s t}$ , in the active $ϕ_{a c}$ and absorbing $ϕ_{a b}$ phases, vs $L$ .
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