Infinitely-many absorbing-state nonequilibrium phase transitions

Abstract

We present a general field-theoretic strategy to analyze three connected families of continuous phase transitions which occur in nonequilibrium steady-states. We focus on transitions taking place between an active state and one absorbing state, when there exist an infinite number of such absorbing states. In such transitions the order parameter is coupled to an auxiliary field. Three situations arise according to whether the auxiliary field is diffusive and conserved, static and conserved, or finally static and not conserved. I The ubiquity of absorbing-state transitionsThis overview is devoted to a study of nonequilibrium phase transitions taking place between the active and the absorbing state of a system, as some control parameter is varied across a threshold value. Such transitions are encountered in a variety of fields ranging from chemical kinetics to the spreading of computer viruses [1]. From a theoretical standpoint absorbing state transitions form natural counterparts to equilibrium phase transitions. The transition rates used in the stochastic dynamics employed to model the physical phenomenon under consideration do not satisfy detailed balance (with respect to an a priori defined energy function). In spite of this apparent freedom, the number of universality classes that the transition can fall into is incredibly small. Among known universality classes, that of directed percolation (DP) is by far the broadest. And indeed, in the absence of additional symmetries or conservation laws, as was conjectured twenty years ago by Grassberger [2], an absorbing state transition will invariably fall into the DP universality class. The interest in absorbing state transitions was further enhanced as Dickman and coworkers [3] established a one-to-one correspondence with self-organized critical systems (see [4] for a review on self-organized criticality). They were able to show that the scaling behavior observed there was entirely governed by an underlying nonequilibrium phase transition (which, as a side effect, somewhat tempers the mystics of SOC). The study of exactly which microscopic ingredients make an absorbing state transition not belong to the DP class has almost grown into a field of its own. It was early realized that if the microscopic dynamics possesses additional conservation laws the universality class of the transition could be different. Discrete conservation laws, such as the conservation of the parity of the number of particles [5], are known to be driving the transition to an independent universality class (the Parity Conserving or Voter class [6]). A recent study attempts to provide a comprehensive table of all possible transitions involving the dynamics of a single field [7]. Besides, the effect of a continuous symmetry was shown either to change the universality class of the transition [8,9] or to simply destroy its continuous nature [10]. The continuous symmetry present in the systems studied there arose from a local conservation law.An independent direction of research has focused on absorbi...

Phase transitions with infinitely many absorbing states in complex networks

“…While simple models such as the CP are characterized by a unique absorbing state devoid of all particles, other relevant systems present multiple absorbing configurations [21]. Examples of those range from models of forest fires [22] to self-organized critical sandpiles [23].…”

Soluble one-dimensional particle conservation models with infinitely many absorbing states

“…from which we may obtain the derivative of λ with respect to h. After some straightforward algebraic steps and taking into account the relation (12) we arrive at the following relation between ρ a and λ:…”

“…Lattice models with infinitely many absorbing states with particle conservation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] are attractive due to their close connection [2,[16][17][18][19]] to self-organized criticality (SOC) [20][21][22]. They are characterized by displaying a continuous phase transition from an absorbing to an active state as one increases the density of particles, which represents a nondiffusive conserved field.…”

“…Above a certain finite critical density of particles the system presents an active state in which the density of active sites is nonzero. The critical behavior places these models into a distinct universality class, namely that of absorbing phase transition with a nondiffusive conserved field [1,3,9,10,12,13].…”

An asymmetric sandpile model with height restriction

“…Self-organized criticality (SOC) has been successfully described by sandpile lattice models [1][2][3]. There is a close connection between SOC and stochastic lattice models with infinitely many absorbing states and a nondiffusive conserved field [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Some of them, which concern us here, are called fixed-energy sandpile models [3,7,9,12,16,18].…”