Based on quasistationary distribution ideas, a general finite size scaling theory is proposed for discontinuous nonequilibrium phase transitions into absorbing states. Analogously to the equilibrium case, we show that quantities such as response functions, cumulants, and equal area probability distributions all scale with the volume, thus allowing proper estimates for the thermodynamic limit. To illustrate these results, five very distinct lattice models displaying nonequilibrium transitions-to single and infinitely many absorbing states-are investigated. The innate difficulties in analyzing absorbing phase transitions are circumvented through quasistationary simulation methods. Our findings (allied to numerical studies in the literature) strongly point to a unifying discontinuous phase transition scaling behavior for equilibrium and this important class of nonequilibrium systems.

Generic finite size scaling for discontinuous nonequilibrium phase transitions into absorbing states.

We study absorbing-state phase transitions (APTs) in two-dimensional Voronoi-Delaunay (VD) random lattices with quenched coordination disorder. Quenched randomness usually changes the criticality and destroys discontinuous transitions in low-dimensional nonequilibrium systems. We performed extensive simulations of the Ziff-Gulari-Barshad model, and verified that the VD disorder does not change the nature of its discontinuous transition. Our results corroborate recent findings of Barghathi and Vojta [H. Barghathi and T. Vojta, Phys. Rev. Lett. 113, 120602 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.120602], stating the irrelevance of topological disorder in a class of random lattices that includes VD, and raise the interesting possibility that disorder in nonequilibrium APT may, under certain conditions, be irrelevant for the phase coexistence. We also verify that the VD disorder is irrelevant for the critical behavior of models belonging to the directed percolation and Manna universality classes.

Continuous and discontinuous absorbing-state phase transitions on Voronoi-Delaunay random lattices.

We survey the effect of perturbing the regular structure of a cellular automaton. We are interested in critical phenomena, i.e., when a continuous variation in the local rules of a cellular automaton triggers a qualitative change of its global behaviour. We focus on three types of perturbations: (a) when the updating is made asynchronous, (b) when the transition rule is made stochastic, (c) when the topological defects are introduced. It is shown that although these perturbations have various effects on CA models, they generally produce the same effects, which are identified as first-order or second-order phase transitions. We present open questions related to this topic and discuss implications on the activity of modelling.

https://hal.inria.fr/inria-00412278/document

Critical phenomena in cellular automata: perturbing the update, the transitions, the topology

We present a detailed study of the prisoner's dilemma game with stochastic modifications on a two-dimensional lattice, in the presence of evolutionary dynamics. By very nature of the rules, the cooperators have incentives to cheat and fear being cheated. They may cheat even when this is not dictated by the evolutionary dynamics. We consider two variants here. In each case, the agents mimic the action (cooperation or defection) in the previous time step of the most successful agent in the neighborhood. But over and above this, the fraction p of cooperators spontaneously change their strategy to pure defector at every time step in the first variant. In the second variant, there are no pure cooperators. All cooperators keep defecting with probability p at every time step. In both cases, the system switches from a coexistence state to an all-defector state for higher values of p. We show that the transition between these states unambiguously belongs to the directed percolation universality class in 2 + 1 dimensions. We also study the local persistence. The persistence exponents obtained are higher than the ones obtained in previous studies, underlining their dependence on details of the dynamics.

http://iopscience.iop.org/article/10.1088/1742-5468/2010/03/P03016/pdf

Dynamic phase transition in the prisoner's dilemma on a lattice with stochastic modifications

How do multiple active components at one level of organisation create agential wholes at higher levels of organisation? For example, in organismic development, how does the multi-scale autonomy of the organism arise from the interactions of the molecules, cells and tissues that an organism contains? And, in the major evolutionary transitions, how does a multicellular organism, for example, arise as an evolutionary unit from the selective interests of its unicellular ancestors? We utilise computational models as a way to think about this general question. We take a deliberately minimalistic notion of an agent: a competency to take one of two possible actions to minimise stress. Helping ourselves to this behaviour at the microscale, we focus on conditions where this same type of agency appears spontaneously at a higher level of organisation. We find that a simple process of positive feedback on the timing of individual responses, loosely analogous to the natural phase synchronisation of weakly coupled oscillators, causes such a transition in behaviour. The emergent collectives that arise become, quite suddenly, able to respond to their external stresses in the same (minimal) sense as the original microscale units. This effects a dramatic rescaling of the system behaviour, and a quantifiable increase in problem-solving competency, serving as a model of how higher-level agency emerges from a pool of lower-level agents or active matter. We discuss how this dynamical ‘waking-up’ of higher-level collectives, through the alignment of their internal dynamics, might relate to reproductive/cell-cycle synchronisation in evolutionary transitions and development.

An ability to respond begins with inner alignment: How phase synchronisation effects transitions to higher levels of agency

The emergence of cooperation in self-centered individuals has been a major puzzle in the study of evolutionary ethics. Reciprocal altruism is one of the explanations put forward and the prisoner's dilemma has been a paradigm in this context. The emergence of cooperation was demonstrated for the prisoner's dilemma on a lattice with synchronous update. However, the cooperation disappeared for asynchronous update and the general validity of the conclusions was questioned. Neither synchronous nor asynchronous updates are realistic for natural systems. In this paper, we make a detailed study of a more realistic system of semi-synchronous updates where pN agents are updated at every time instant. We observe a transition from an all-defector state to a mixed state as a function of p. Our studies indicate that despite it being a transition from an absorbing state, it is a first-order transition. Furthermore, we used a damage spreading technique to demonstrate that the transition in this system could be classified as a frozen–chaotic transition.

The prisoner's dilemma with semi-synchronous updates: evidence for a first-order phase transition

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The prisoner's dilemma with semi-synchronous updates: evidence for a first-order phase transition