A class of exactly solved assisted-hopping models of active-inactive state transition on a line

Abstract

We construct a class of assisted hopping models in one dimension in which a particle can move only if it does not lie in an otherwise empty interval of length greater than n + 1. We determine the exact steady state by a mapping to a gas of defects with only on-site interaction. We show that this system undergoes a phase transition as a function of the density ρ of particles, from a low-density phase with all particles immobile for ρ ≤ ρc = 1 n+1, to an active state for ρ > ρc. The mean fraction of movable particles in the active steady state varies as (ρ − ρc) β , for ρ near ρc. We show that for the model with range n, the exponent β = n, and thus can be made arbitrarily large. Introduction -There has recently been a lot of interest in models showing an active-absorbing state transition [1][2][3]. The best studied of these is the directed percolation (DP) class [4,5], and the related pair-contact process with diffusion (PCPD) [6][7][8] and parity-conserving directed percolation (PC) [9,10]. Models with conserved number of particles where a particle may hop only if there are sufficient number of other particles present within a given range, also show an active-absorbing phase transition, where the lowdensity phase is inactive. Examples of this class are the conserved lattice gas model and the conserved threshold transfer process[11], fixed energy sandpiles [12] and the activated random walkers model [13]. By adding a slow drive and dissipation, these models can be converted into models showing self-organized criticality [14]. It has recently been argued that on adding symmetry-breaking perturbations, these models would flow to DP-like critical behavior, but the evidence for this scenario is mainly numerical [15,16]. It seems worthwhile to investigate further models with conserved particle number showing active-absorbing phase transitions.

Exclusion processes with avalanches

“…Equation (33) can be rewritten as the diffusion equation (24) with D(ρ) = dR dρ . Combining this relation with (34) we arrive at the announced diffusion coefficient (25).…”

Multichain models of conserved lattice gas

“…APT in presence of a conserved field [23,24] has been a subject of interest in recent years. The conserved lattice gas (CLG) model [25,26] and some of its extensions [27,28] are exactly solvable in one dimension * arijit.chatterjee@saha.ac.in † pk.mohanty@saha.ac.in and they provide clear examples of non-DP behaviour. These models are rather simple having trivial integer exponents.…”

“…These models are rather simple having trivial integer exponents. Some variations of CLG models also show continuously varying critical exponents or multi-critical behaviour [27,28]. Non-DP behaviour in these models can not be blamed to presence of the conserved density because the same dynamics on a ladder geometry lead to an absorbing transition belonging to DP class [21].…”

Exact hyperuniformity exponents and entropy cusps in models of active-absorbing transition

“…In this letter, we study two models with a conserved density that show an active-absorbing phase transition in one dimension. The first model is the abelian sandpile model (ASM), while the second is the class of conserved lattice gases with extended range (CLGs) whose the active steady state behaviour was exactly solved in [8]. We show that the introduction of a quasistatic drive leads to an exactly solvable measure on the absorbing states in these models.…”

“…1D Conserved Lattice Gases with extended range Conserved Lattice gases are models of particles on the lattice with exclusion interaction, and the property that particles can only move if another particle is present within a specified range. The class of conserved lattice gases we study in this section was defined in [8], where the properties of the active steady state were characterised exactly. Consider N particles on L sites on a 1D ring, such that each site can have no more than 1 particle.…”